Thread: 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

Originally Posted by tttcomrader

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,

Is called the Basel Problem.
It was first solve (non-rigorusly) by Euler.

Basically what this says is that the sequence of partial sums,



....
Has a least upper bound which is .

Thus, if represents the n-th partial sum we have (since it is an upper bound) that,

Subtract from both sides,

Which is true for .

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