$p$-Series and the Exact Value of $$\sum_{n=1}^{\infty}\frac{1}{n^2}$$

The term $p$-series refers to series of the form $$\sum_{n=1}^{\infty} \frac{1}{n^p} = \frac{1}{1^p} +\frac{1}{2^p} +\frac{1}{3^p}+\cdots$$. It can be easily shown in calculus, using the integral test, that the $p$-series diverges for $p\le1$ and converges for $p>1$. We have already seen that $p=1$ leads to a divergent series. Now we will finally investigate the Basel Problem, the $p$-series for $p = 2$.

1. Since $$\frac{1}{n^2}<\frac{2}{n(n+1)}$$, show by using a telescoping sum how Jakob Bernoulli was able to prove that $$\sum_{n=1}^{\infty}\frac{1}{n^2}<2$$
2. Euler showed that $$\sum_{n=1}^{\infty}\frac{1}{n^2}=\frac{\pi^2}{6}$$.
   1. Find the exact sum of $\frac{1}{15}+\frac{1}{63}+\frac{1}{80}+\frac{1}{255}+\frac{1}{624}+\cdots$. ( The reciprocals of the numbers that are one less than perfect squares which simultaneously are other powers, i.e. $16= 4^2=2^4$ so $16-1=15$ is a denominator in the series.)
   2. Find the ratio of $$\sum_{n=1}^{\infty} \frac{1}{n^p}$$ to $$\sum_{n=1}^{\infty} \frac{(-1)^{n+1}}{n^p}$$ , where $p>1$.
   3. Find $$\sum_{n=1}^{\infty} \frac{1}{(2n-1)^2}$$.
   4. Find $$\sum_{n=1}^{\infty} \frac{1}{n^4}$$.