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May you help me with this problem?

Thank you so much!

How can I prove that series number 1 is equal to series number 2?

I noticed that series number 1 is equal to $\frac{1}{k^4}$ when $k$ is even and to $\frac{2}{k^4}$ when $k$ is odd. But, no more than that...

Here is the series number 1

$$\large{\sum_{k=1}^{\infty} \frac{1-\cos\left(\frac{k\pi}{2}\right)}{k^4}}$$

and here is the series number 2

$$\large{\sum_{k=1}^{\infty} \left(\frac{1}{k^4}+\frac{(-1)^{k-1}}{(2k)^4}\right)}$$

  • 1
    You might want to check "I noticed that series number **1** is equal to $\frac{1}{k^4}$ when $k$ is even and to $\frac{2}{k^4}$ when $k$ is odd" – 2017-02-26
  • 0
    @Henry Thank you for your help! – 2017-02-26

1 Answers 1

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Note that $ \cos(j\pi/2)$ is zero if $j$ is odd and $(-1)^k$ if $j$ is even $j=2k$. The zero terms have been removed & then the sum has been reordered to give the expression stated in your question.

  • 2
    +1. Also worth noting that $\displaystyle\sum \frac1{k^4}$ and similar series are absolutely convergent, thus allowing reordering – 2017-02-26