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Numerade Educator

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Problem 34 Medium Difficulty

Leonhard Euler was able to calculate the exact sum of the $ p- $ series with $ p = 2: $
$ \zeta (2) = \displaystyle \sum_{n = 1}^{\infty} \frac {1}{n^2} = \frac {\pi^2}{6} $
(See page 720.) Use this fact to find the sum of each series.
(a) $ \displaystyle \sum_{n = 2}^{\infty} \frac {1}{n^2} $
(b) $ \displaystyle \sum_{n = 3}^{\infty} \frac {1}{(n + 1)^2} $
(c) $ \displaystyle \sum_{n = 1}^{\infty} \frac {1}{(2n)^2} $

Answer

a) $\frac{\pi^{2}}{6}-1$
b) $\frac{6 \pi^{2}-49}{36}$
c) $\frac{\pi^{2}}{24}$

Discussion

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Tamah T.

November 4, 2020

why did we subtract 1/1 and not another number say, 1/n^2?

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Tamah T.

November 4, 2020

LN

Letisia N.

May 3, 2021

In 1795, Leonhard Euler proved that the sum of the following infinite series is equal to ?2=6 1 X k =1 1 2k = 1 + 1 22 + 1 23 + 1 24 + · · · (1) a). Use a for loop to sum the first twenty terms of the series and compare this partial sum with ?2=6 (compute

LN

Letisia N.

May 3, 2021

Video Transcript

for part A should realize that this is the sum that we know. But just one term father and silence dead right the sun from one to infinity and then subtract the n equals one term which should be one over one. And since we know what the first number is, it's pi squared over six. Forget pi squared over six minus one part B. It's ah, little bit trickier. There's, ah, stuff we want to do we have We know what one over and squared is but we don't know one over and plus one squared is however one over and plus one squared is just one over end squared but starting turn back so we can replace that could only with some from Ann equals four to infinity I've just won over and squared Uh, and this will end up equaling. Hi scored over six, which is the one to infinity term minus the first three terms one one fourth and one night which simplifies too. Pi squared over six minus forty nine over thirty For the final sum, we should apply that squared the inside So one over to n squared is the same thing is one over four and squared, which, if we pull the one fourth out, it's just one fourth times some that we already know that'LL be one fourth times pi squared over six or pi squared over twenty four.