# Thread: Induction on a sequence

1. ## Induction on a sequence

Q: Use the principle of induction to prove that 1/1^2 + 1/2^2 + ... + 1/n^2 <= 2 - 1/n for all n within the set of Natural Numbers.

My Solution: Let S = {n within N : {1/n^2} <= 2 - 1/n}

1 is within N, and 1/1^2 <= 2 - 1/1 = 1, so 1 is within S.
Let k be within S.

Proof: (k+1) is within S...

Any hints?

KK

Q: Use the principle of induction to prove that 1/1^2 + 1/2^2 + ... + 1/n^2 <= 2 - 1/n for all n within the set of Natural Numbers.

My Solution: Let S = {n within N : {1/n^2} <= 2 - 1/n}

1 is within N, and 1/1^2 <= 2 - 1/1 = 1, so 1 is within S.
Let k be within S.

Proof: (k+1) is within S...

Any hints?

KK
The infinite series,
$\displaystyle \sum_{k=1}^{\infty}=1+\frac{1}{2^2}+\frac{1}{3^2}+ ...$
Is called the Basel Problem.
It was first solve (non-rigorusly) by Euler.

Basically what this says is that the sequence of partial sums,
$\displaystyle 1$
$\displaystyle 1+\frac{1}{2^2}$
$\displaystyle 1+\frac{1}{2^2}+\frac{1}{3^2}$
....
Has a least upper bound which is $\displaystyle \frac{\pi^2}{6}$.

Thus, if $\displaystyle H_n$ represents the n-th partial sum we have (since it is an upper bound) that,
$\displaystyle H_n\leq \frac{\pi^2}{6}$
Subtract $\displaystyle 1$ from both sides,
$\displaystyle \frac{1}{2^2}+\frac{1}{3^2}+...+\frac{1}{n^2}\leq \frac{\pi^2}{6}\approx 1.64<2-1/n$
Which is true for $\displaystyle n\geq 3$.

Note, if you wish I can show you how to prove the Basel sum using a Fourier series?
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