Squared harmonic series

Calculus Level 1

The harmonic series $\frac { 1 }{ 1 } +\frac { 1 }{ 2 } +\frac { 1 }{ 3 } +\frac { 1 }{ 4 }+ \cdots$ appears to approach a single number as the number of terms increases. However, as the number of terms approaches infinity, the sum also approaches infinity!

Based on this information, does the sum $\frac { 1 }{ { 1 }^{ 2 } } +\frac { 1 }{ { 2 }^{ 2 } } +\frac { 1 }{ { 3 }^{ 2 } } +\frac { 1 }{ { 4 }^{ 2 } } +\cdots$ also approach infinity as the number of terms approaches infinity?

(Challenge: Solve this problem WITHOUT using the riemann zeta function.)

Yes No

3 solutions

Tijmen Veltman
Feb 27, 2018

Without Riemann-Zeta:

$1/1^2 + 1/2^2+1/3^2 + 1/4^2+1/5^2+1/6^2+1/7^2+ \ldots$

$= 1/1^2 + (1/2^2+1/3^2) + (1/4^2+1/5^2+1/6^2+1/7^2)+ \ldots$

$< 1/1^2 + (1/2^2+1/2^2) + (1/4^2+1/4^2+1/4^2+1/4^2)+\ldots$

$= 1/1^2 + 2/2^2 + 4/4^2 + \ldots$

$= 1 + 1/2 + 1/4 + \ldots$

$=2$

Therefore this sum will certainly not approach $+\infty$ .

It should be 4/4^2

Tobias Rupp - 2 years, 9 months ago

Good catch, I edited it!

Tijmen Veltman - 2 years, 9 months ago

Suresh Jh
Feb 18, 2018

With riemann zeta function sum = $\frac{π^2}{6}$ , so it number not trends to infinite.

Giorgos K.
Feb 18, 2018

It approaches π^2/6
see details here
http://mathworld.wolfram.com/RiemannZetaFunctionZeta2.html

Squared harmonic series

3 solutions

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