High school Calculus homework on series

In summary, the conversation discusses finding a series for various functions and using the result to approximate pi. The first step is to find a series for f(x) = 1/(1+x) and determine the interval of convergence. This can be done by using the formula -1 < x < 1. Then, the result from 1) is used to find a series for 1/(1+x^2) in 2). Finally, the series from 2) is used to approximate pi in 3) by replacing x with x^2 and using the derivative of tan-1x. The number of terms needed for a four decimal place approximation is not specified.
  • #1
snowlove
2
0
high school Calculus B/C class homework:

1) Find a Series for f(x) = [tex]\frac{1}{1+x}[/tex] then find the interval of convergence.

i know that [tex]\frac{1}{1+x}[/tex] = 1-x+ [tex]x^{2}[/tex]-[tex]x^{3}[/tex]+...+ [tex]-x^{n}[/tex]+... -1<x<1
but then i don`t understand how can i use the result from 1) to find 2) answer? since you guy will teach me how to do 2) then do i just do the same thing for 3) ?

2) Using your result from 1) find a series for [tex]\frac{1}{1+x^2}[/tex]

3) Using your result from 2) find a series for [tex]tan^{-1}x[/tex] then use your series to approximate [tex]\pi[/tex] to four decimal places. How many terms did you need?
 
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  • #2


Replace x by x2

For the third one, consider what d/dx(tan-1x) gives
 
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  • #3


thank you
 

1. What is a series in calculus?

A series in calculus is a sum of infinitely many terms. It is represented by sigma notation, where the expression for each term is written inside the sigma symbol with the starting and ending values of the index shown below and above the symbol, respectively.

2. How do I find the sum of a series?

To find the sum of a series, you can use various techniques such as the geometric series formula, the telescoping series method, or the ratio test. It is important to determine if the series is convergent or divergent before attempting to find the sum.

3. What is the difference between a convergent and divergent series?

A convergent series is one in which the terms of the series approach a finite limit as the number of terms increases. In other words, the sum of a convergent series is a finite number. On the other hand, a divergent series is one in which the terms of the series do not approach a finite limit and the sum of the series is infinity.

4. How do I determine if a series is convergent or divergent?

There are various tests that can be used to determine the convergence or divergence of a series, such as the divergence test, integral test, comparison test, and root test. It is important to understand the conditions for each test and apply them appropriately to the given series.

5. Can I use a calculator to solve series problems?

While calculators can be useful for checking calculations, it is important to understand the concepts and techniques involved in solving series problems by hand. Using a calculator to solve series problems can also be limited as it cannot always provide explanations for the steps involved in the solution process.

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