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English Guide: Practical Tips for Better English Communication

thank you We all have certain people in our life to whom we are really thankful. It may be because they might have helped us or have been there when we needed their support.To show our courtesy, we must thank them in a good way. People give chocolates, gifts etc. to show the gratitude. The simplest form of expressing gratitude is by using the kind words. So, let us have a look at some of the best ways to thank someone. 8 Interesting ways to thank someone1....
sad baby In life, there are times when we feel low and upset. This situation comes when some unexpected things happen. We all, for sure, might have faced this kind of situation at least once in our lifetime. However, these things are very common and expressing them in English is not uncommon. In this blog, you will learn about some interesting slang words to express and describe disappointment.8 slang words to describe disappointment1. Bummer or Bummer...
Jam session In today’s competitive and busy world, it is not easy to get a job in an MNC company. As there are many candidates who apply on a daily basis, it becomes difficult for the HR to conduct the interviews. Hence, they conduct various rounds and the one who clears reaches the final round. These rounds include group discussions, written tests, and a JAM session. While you might be aware of tests and group discussions, JAM can be described as a short...
pronounce words English is the only language which is spoken widely all over the world as it can be easily learnt and understood. One of the old yet easiest way to learn English is by reading newspapers, books, novels etc. You might find range of vocabularies to learn. Pick up 10 new words and learn the meaning, spelling, pronunciation etc. Unfortunately, there are many words which are pronounced incorrectly. According to a recent research almost 82% of the people...
travel idiom An idiom is an expression, word or a phrase that has a figurative meaning. They don’t mean exactly what the word say. They have hidden meaning. Like for example, “it’s raining cats and dogs”. The sentence doesn’t mean the way it is said or written. However, the sentence means it is raining heavily. In this blog, we will discuss travel idioms.Travel Idioms one must know1. Asleep at the wheel: Not attentiveExamples:Yash scored less marks in the final exams because...
travel idiom In the previous blog, we explained some interesting travel idioms. In today’s blog, we will explain some more travel idioms that will help to improve English communication skills. Let us learn those travel idioms.8 Travel Idioms you must know1. Road rage: anger or violence towards other drivers.Examples:While returning from Pune, our driver had a road rage with the other driver.Earlier today a man was arrested for attacking a motorist in a road...
phrasal verb with do Phrasal verbs are a combination of a verb and preposition, a verb, and an adverb. They are also known as ‘compound verb’. For e.g. - Don’t put me off, I am trying my best. There are many phrasal verbs which we use in our day to day communication. In today’s blog, we will explain phrasal verbs using the word “do”.Phrasal verbs using do:1. Do up: make something look good.Examples: I will help you to do up your hair.You should try doing up...
bossy   In the professional world, we use a formal English for communicating with our colleagues. But sometimes bosses and managers use expressions which you have never heard about. In today’s blog, we are bringing you bossy words which you need to understand in order to be comfortable and stable in the professional world.8 Bossy words you must know:1. Under my nose: to do some work under someone’s supervision.Examples: The new boss is so strict that...
idiom world In your day to day life you meet many people with different behaviours and attitudes. Some are short-tempered, some are down to earth while some are shy and others are talkative. While maintaining friendship with them, you might have surely faced a situation where you had to stop someone from talking  rubbish things. Situations like these need to be handled calmly without actually offending the person. In this blog, we bring you idioms that can...
homophone Have you heard these words- ad, add; bare, bear; been, bean and got confused with the spellings? These words are pronounced in a similar way but are spelled differently and mean different. These words are known as homophones. There are some more examples of homophones. So, let us have a look at some more interesting homophones. 7 interesting homophones:1. Allowed and AloudAllowed means when you get a permission from someone to do...
socializing in English In our day to day life, we use many words to converse in English. Many of these words used are a combination of a verb and a preposition or a combination of verb and adverb. These words are called Phrasal verbs. These words give an entirely different meaning when combined. It is vital to know their proper meaning in order to use them in a sentence. Till now we have listed many blogs related to phrasal verbs and its usage to describe your day....
car All these years we published blogs that were related to interview skills or English grammar. Just a few days back we started creating more blogs that helped the beginners to learn basic English skills. The basic English skills include vocabularies, phrases, idioms that are used in day to day communication. While the blogs provided information on general topics, today’s blog is for those who love cars, bikes etc. i.e. for men and boys. However,...
funny idiom to improve English 1 In our last blog, we described funny idioms that are often used by the native speakers in the day to day communication. Funny idioms not only add a humorous touch but also make you a pro-English speaker. Let us have a look at the remaining idioms that are funnier than the previous one. 6 Funny idioms you must know1. Chew the fatThis idiom means to chat in a friendly and leisurely way or engage in casual gossip sessions. It is generally said to...
funny idiom to improve English Idioms are an everyday figure of speech which is taken in a light and a humorous way. The figurative meaning of the idioms makes the sentence often go against the logical “rules of language and grammar”.  Native speakers use these idioms daily in the conversation which not only improves English but also helps to get the command over the language. In today’s blog, we’ll teach funny idioms which will help improve your English-speaking skills.5...
confused In life, we have faced many instances where we have misunderstood many things and have ended up confused. We were able to clear a few, while there was a lot of confusion in the others. It is important to clarify and clear out the confusion in order to avoid unexpected results. To understand and clarify the confusion, you are required to articulate the points in an impeccable English with good vocabulary words. Today’s blog will help you with...
food “Food is love food is life” for all the food lovers. It’s hard to define the relation that the food lovers share with food. While people explore cuisines from different places to get a sense of taste, they also learn many new things. Learning new things is easy and so is English. As a foodie, you learn new dishes and its ingredients which indirectly help you to learn new words, spellings and pronunciation. Sometimes describing them is difficult...
food vocabulary In our last blog, we described vocabulary related to texture of the food. This blog will introduce more words about food that can be used while communicating. Food vocabulary is useful for everyone – right from the housewives to the food lovers. Have you ever been in the phase where you found it difficult to explain a recipe in English? If your answer is yes, then don’t worry. It’s something that everybody has gone through. There are many words that...
conversation starters When you go to a new place, where you don’t know anyone, it becomes a bit difficult to spend the time. In order to know more about the festival, culture, climate etc. of that place you decide to explore the city. You find a group of locals there and decide to make friendship and know about the city. A fluent English, in this case, is very helpful to start the conversation. You need to be aware of the conversation starters that will encourage...
grammar mistakes Are you a beginner in English? It is common to get confused with similar words or phrases. Speaking English is easy as it just takes confidence and a person to talk with. Don’t be afraid of mistakes, mistakes are the proof that you are trying. Only by making mistake you can learn new things. It is not that you are the only one who is making mistakes. There are many native English speakers who get confused while speaking. The mistakes can be...
teen slang Slang words are informal language words that are used by the teens and youngsters. Today, it’s impossible to be in a nightclub, or at a cutting-edge art gallery, without hearing these slang words. These are nothing but idioms and expressions, which make adults scratch their head. Don’t worry, this blog will explain you slang words along with its meaning.10 slang words you must know1. It sucks / that sucksMeaning - It might not be the right...
misused words literally In our last blog, we discussed words that are commonly misused in English. In this article, we will add some more words that are commonly misused. Knowing the correct usage will help you to enhance your vocabulary as well as spoken English Skills which will help you to speak effectively. Let us look at the words.10 words with incorrect usage1. An effectAn effect means either to influence or to fake whereas to effect means to put into effect.Examples:The new...
you can do it In life, there are many situations which make us sad and feel low. At that moment when someone encourages us with their positive words, we feel light and happy.  Don’t we? We encourage people playing sports, while building relationships with friends, family, colleagues, and while working. In the same manner, when your friend or your loved one feels low, you try to encourage them with positive words by complimenting and appreciating them. In...
give it a shot In our last blog, we taught you few phrases that can be used for encouraging someone. When you use these phrases in your communication, it not only helps you to motivate someone but also improves your English-speaking skills. Let’s have a look at some positive phrases that will improve your English skills and help you encourage someone.6 phrases you must know1. You are almost there!This phrase is best to encourage someone who is about to...
interjections In our earlier blog, we provided a fun guide to interjections which help us in our day to day conversation. Interjections are words for the emotions and expressions that we make. These interjections words depict our emotions like surprise, agreement, disagreement, sarcasm, shock etc. In today’s blog, we will explain interjections to be used to express your emotions which will help you in your everyday conversations using “expressive...
couch potato Idioms play an important role in English language. They help to express your feelings in a figurative manner. In today’s lesson, we will describe people and their characteristics with idiomatic expressions. These people might be your friends, cousins, co-workers and to some your family members. All these characters are different from each other and have a unique differentiating factor through which we identify or know them. Let us learn the...

TAYLOR SERIES

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Introduction

taylorseries1

If ‘f’ has derivatives of all orders at x0, then we call the series

taylorseries2

The Taylor series for ‘f’ about x=x0
In the special case where x0=0, this series becomes

taylorseries3

which is called as the Maclaurin series of ‘f’.
Note that the nth Maclaurin and Taylor polynomials are the nth partial sums for the corresponding Maclaurin and Taylor series.
Example: Find the Maclaurin series for 1/(1-x)
Solution: We know that the nth Maclaurin polynomial for 1/(1-x) is
pn(x) = Σxk = 1+x+x2+………+xk (n=0, 1, 2………….)
Thus, the Maclaurin series for 1/(1-x) is
Σxk= 1+x+x2+x3+…….+xk+…………..
Example: Find the Taylor series for 1/x about x=1
Solution: We know that the nth Taylor polynomial for 1/x about x=1 is
Σ(-1)k(x-1)k=1-(x-1)+(x-1)2-(x-1)3+….+(-1)n(x-1)n
Thus the Taylor series for 1/x about x=1 is
Σ(-1)k(x-1)k=1-(x-1)+(x-1)2-(x-1)3+……..+(-1)k(x-1)k+……….

Power Series in x

Maclaurin and Taylor series differ from the series that we have discussed above that their terms are not merely constants, but instead involve a variable.  These are examples of power series which is defined as
If c0, c1, c2, ……. Are constants and x is a variable then a series of the form
        Σckxk= c0+c1x+c2x2+………..+ckxk+……………….. is called a power series in x
Here are some examples
taylorseries4
Indeed every Maclaurin series is a power series in x.


Radius and interval of convergence

If a numerical value is substituted for x in a power series Σckxk, then the resulting series of numbers may either converge or diverge.  This leads to the problem of determining the set of x-values for which a given power series converges; this is called its convergence set.
For any a power series in x, exactly one of the following is true:
a)    The series converges only for x=0
b)    The series converges absolutely (and hence converges) for all real values of x
c)    The series converges absolutely (and hence converges) for all x in some finite open interval (-R, R) and diverge if x<-R or x>R.  At either of the values x=R or x=-R, the series may converge absolutely, converge conditionally or diverge, depending on the particular series.
This theorem states that the convergence set for a power series in x is always as interval centered at x=0 (possibly just the value x=0 itself or possibly infinite).  For this reason, the convergence set of a power series in x is called the interval of convergence.  In the case where the convergence set is the single value x=0 we say that the series has radius of convergence 0, in the case where the convergence set is (-∞, +∞) we say that the series has radius of convergence +∞ and in the case where the convergence set extends between –R and R we say that the series has radius of convergence R.    

taylorseries5

The usual procedure for finding the interval of convergence of a power series is to apply the ratio test for absolute convergence.  The following example illustrates how this works.
Example: Find the interval of convergence and radius of convergence of the following power series Σxk and Σxk/k!

Solution:
Σxk
We apply the ratio test for absolute convergence.  We have
             taylorseries6
So the series converges absolutely if ρ = |x| <1 and diverges if ρ=|x|>1.  The test is inconclusive if |x|=1 (x=1 or x=-1), which means that we will have to investigate convergence at these values separately.  At these values the series becomes
               Σ1k = 1 + 1+ 1+ 1+ 1+………… [x=1]
               Σ(-1)k = 1-1+1-1+1-1+……..[x=-1]
Both of which diverge, thus, the interval of convergence for the given power series is (-1, 1) and the radius of convergence is R=1.

Σxk/k!
Applying the ratio test for absolute convergence, we obtain
   taylorseries7


Power Series in x-x0

Σck(x-x0)k = c0+c1(x-x0)+c2(x-x0)2+…..+ck(x-x0)k+…… is called a power series in x-x0.
For a power series Σck(x-x0)k, exactly one of the following statements is true:
a)    The series converges only for x=x0.
b)    The series converges absolutely and hence converges for all real values of x.
c)    The series converges and hence converges for all x in some finite open interval (x0-R, x0+R) and diverges is x<x0-R or x>x0+R.  At either of the values x=x0-R or x=x0+R, the series may converge absolutely, converge conditionally or diverge depending on the particular series.
It follows from the above statements that the set of values for which a power series in x-x0 converges is always an interval centered at x=x0; we call this the interval of convergence.  In part (a) the interval of convergence reduces to the single value x=x0 in which case we say that the series has radius of convergence R=0; in part (b) the interval of convergence is infinite, in which case we say that the series has radius of convergence R=+∞; and in part (c) the interval extends between x0-R and x0+R, in which case we say that the series has radius of convergence R.
                                                                      
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