Obtain the sum of the series

$\sum _{n\=1}^{\infty }\frac{{\left(-1\right)}^{n\+1}}{n^2}$ = $\frac{1}{12}{\mathrm{\π}}^2$$$

Obtain the partial sum $S_k$ and the first few values of $S_k$ 

$\mathrm{S\_\_k}≔\sum _{n\=1}^k\frac{{\left(-1\right)}^{n\+1}}{n^2}\:$

$\mathrm{value}\left(\mathrm{S\_\_k}\right)$

$\frac{1}{12}\frac{12\mathrm{LerchPhi}\left(-1\,2\,k\right){\left(-1\right)}^k{k}^2\+{\mathrm{\π}}^2{k}^2-12{\left(-1\right)}^k}{k^2}$

Obtain the first few values of $S_k$ 

$\mathrm{value}\left(\left[\mathrm{seq}\left(\mathrm{S\_\_k}\,k\=1..10\right)\right]\right)$

$\left[1\,\frac{3}{4}\,\frac{31}{36}\,\frac{115}{144}\,\frac{3019}{3600}\,\frac{973}{1200}\,\frac{48877}{58800}\,\frac{191833}{235200}\,\frac{5257891}{6350400}\,\frac{5194387}{6350400}\right]$

Obtain the limit of the sequence of partial sums

$\underset{k\to \infty }{lim}\mathrm{S\_\_k}$ = $\frac{1}{12}{\mathrm{\π}}^2$$$$$

Figure 8.2.3(a)   Convergence of $S_k$ to $S\doteq 0.82247$