Sums 1

Calculus Level 3

Find the value of $\displaystyle \sum_{n=1}^\infty\frac {6n}{n^4} .$

Give your answer to 1 decimal place.

The answer is 7.2.

2 solutions

Akshay Yadav
Dec 8, 2015

First of all simplification,

$S=\displaystyle\sum_{n=1}^{\infty} \frac{6n}{n^{4}}$

$S=6\displaystyle\sum_{n=1}^{\infty} \frac{1}{n^{3}}$

We know,

$\displaystyle\sum_{n=1}^{\infty} \frac{1}{n^{3}}=1.21$

Hence,

$S=6\times1.21$

$S=7.26$

How you knew that $\displaystyle\sum_{n=1}^{\infty} \frac{1}{n^{3}}=1.21$ ?

Akshat Sharda - 5 years, 6 months ago

We use here the riemman zeta function.

Imad Fatimy - 5 years, 6 months ago

Pulkit Gupta
Dec 7, 2015

The general term is 6 / $n^3$ .

$\Sigma$ 1 / $n^3$ is $\zeta$ (3) = 1.20 ( approx). The answer is obtained by multiplying this value obtained by 6 which is clear by looking at the general term.

The main point in here is to use the Riemman zeta function. :) Cheers!

Imad Fatimy - 5 years, 6 months ago

Sums 1

The answer is 7.2.

2 solutions

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