Question

In: Advanced Math

Prove by induction: 1 + 1/4 + 1/9 +⋯+ 1/?^2 < 2 − 1/?, for all...

Prove by induction:

1 + 1/4 + 1/9 +⋯+ 1/?^2 < 2 − 1/?,

for all integers ?>1

Expert Solution

We will use induction on to prove that , for .

Base Step : For ,

As ,

So the statement is true for .

Induction Hypothesis : Suppose the statement is true for that is ,

Induction Step : For ,

, Using induction Hypothesis .

, as .

So , .

That is the statement is true for if we assume it is true for . Also the statement is true for . Hence by induction on the statement is true for all or for all .

Hence , for .

.

If you have doubt or need more clarification at any step please comment .

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Colby Messinger answered 3 months ago

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