Ibirari By Insigamigani 14.pdf

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Ibirari By Insigamigani 14.pdf

 

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14. Ibirari by’Insigamigani. 1. 14. 02. 2006. 2. Insigamigani.. 2.

. 14. INGORO Y’UMURAGE W’U RWANDA: IBIRARI BY’INSI GAMIGANI.

. 14. 2. 3. 4. 13. Amakuru, Politiki, Ubukungu, Diaspora, Imikino, Muzika & Videos. 14. 15.. 14. (14) Insigamigani.

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Ibirari By Insigamigani 14.png

Category:Groupe Dessechez languages
Category:Languages of RwandaQ:

Harmonic series sum to $\pi$

The Sum of harmonic series is:
$$H_n=\sum_{i=1}^n\frac1i$$
How to prove that
$$H_n=\frac{\pi^2}{6}+O(n)$$
How to prove that
$$\sum_{n=1}^\infty\frac1{n\sin^2(n\pi/2)}=\frac{\pi^2}{6}$$
Is there a general method for this type of problem?

A:

Hint: We have
$$\sum_{i=1}^n \frac{1}{i} = \left(\log(n) + \gamma + \frac{1}{2n} + \frac{1}{12n^2} + \ldots \right) + O(1)$$
where $\gamma$ is Euler-Mascheroni’s constant.

A:

We have $$\sum_{n=1}^{\infty} \frac1{n\sin^2(\frac{n\pi}{2})}=\sum_{n=1}^{\infty} \frac1{n\cdot \frac{n\pi^2}{4}}=\sum_{n=1}^{\infty} \frac1{n\cdot \frac{n\pi^2}{4}}=\sum_{n=1}^{\infty} \frac1{n\cdot \frac{n^3}{4\pi^2}}=\sum_{n=1}^{\infty} \frac1{n^3} \frac1{4\pi^2/n^3}$$
$$= \sum_{n=1}^{\infty} \frac1{n^3} \left(\frac{\pi^2}{6}-\frac1{12n^2}+O\left(\frac1{n^4}\right)\right)$$
Can you take it from here?

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