Find a formula for the general term anan (not the partial sum) of the infinite series. Assume the infinite series begins at n=1. frac{2}{1^2+1}+frac{1}{2^2+1}+frac{2}{3^2+1}+frac{1}{4^2+1}+...

Question

Find a formula for the general term anan (not the partial sum) of the infinite series. Assume the infinite series begins at n=1.

\frac{2}{1^{2} + 1} + \frac{1}{2^{2} + 1} + \frac{2}{3^{2} + 1} + \frac{1}{4^{2} + 1} + . . .

Answer 1

In the given infinite series, find the term of the series

a_{1}, a_{2} \dots and a_{n}

The terms of the series is given as:

a_{1} = \frac{2}{1^{2} + 1}

a_{2} = \frac{1}{2^{2} + 1}

a_{3} = \frac{2}{3^{2} + 1}

Here we can see that denominator of each term of the series is represented as

(n^{2} + 1)

:
The numerator of the series is the repetition of 2,1,2,1…
Let the terms of the series is calculated as:
when n is odd then k=2
when n is even then k=1

Answer 2

Jeffrey Jordon

Answered 2021-12-16 Author has 2064 answers

Answer is given below (on video)

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Find a formula for the general term anan (not the partial sum) of the infinite series. Assume the infinite series begins at n=1. frac{2}{1^2+1}+frac{1}{2^2+1}+frac{2}{3^2+1}+frac{1}{4^2+1}+...

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