Linked Questions

2
votes
5answers
327 views

Help with proof using induction: $1 + \frac{1}{4} + \frac{1}{9}+\cdots+\frac{1}{n^2}\leq 2-\frac{1}{n}$ [duplicate]

I am having trouble with the following proof: For every positive integer $n$: $$1 + \frac{1}{4} + \frac{1}{9}+\cdots+\frac{1}{n^2}\leq 2-\frac{1}{n}$$ My work: I have tried to add $\frac{1}{(k+1)^...
3
votes
3answers
194 views

Proof verification for proving $\forall n \ge 2, 1 + \frac1{2^2} + \frac1{3^2} + \cdots + \frac1{n^2} < 2 − \frac1n$ by induction [duplicate]

Prove by mathematical induction: $\forall n \ge 2, 1 + \frac1{2^2} + \frac1{3^2} + \cdots + \frac1{n^2} < 2 − \frac1n$ Basis Step: (We want to show, $P(2)$, which is 1 + $\frac1{2^2}<2-\frac12$)...
3
votes
3answers
145 views

Proof of $\sum_{k = 1}^{n} \frac{1}{k^{2}} < 2 - \frac{1}{n}$ [duplicate]

Prove that for $n\geq 2, \: \sum_{k = 1}^{n} \frac{1}{k^{2}} < 2 - \frac{1}{n} $ I used induction and I compared the LHS and the RHS but I'm getting an incorrect inequality
2
votes
2answers
67 views

Proving that $\sum_{i=1}^n\frac{1}{i^2}<2-\frac1n$ for $n>1$ by induction [duplicate]

Prove by induction that $1 + \frac {1}{4} + \frac {1}{9} + ... +\frac {1}{n^2} < 2 - \frac{1}{n}$ for all $n>1$ I got up to using the inductive hypothesis to prove that $P(n+1)$ is true but I ...
7
votes
4answers
2k views

What is the formula for $\frac{1}{1\cdot 2}+\frac{1}{2\cdot 3}+\frac{1}{3\cdot 4}+\cdots +\frac{1}{n(n+1)}$

How can I find the formula for the following equation? $$\frac{1}{1\cdot 2}+\frac{1}{2\cdot 3}+\frac{1}{3\cdot 4}+\cdots +\frac{1}{n(n+1)}$$ More importantly, how would you approach finding the ...
3
votes
4answers
9k views

Sum of series question: $S_n = 1 + 1/4 + 1/9 + 1/16 + 1/25 + … + 1/n^2 < 2$

Prove that for any nonzero natural $n$ it is true that $$S_n = 1 + 1/4 + 1/9 + 1/16 + 1/25 + … + 1/n^2 < 2.$$ I'm sort of at a loss here. I'm not sure if there exists some formula or method to ...
6
votes
5answers
449 views

Proving inequality for induction proof: $\frac1{(n+1)^2} + \frac1{n+1} < \frac1n$

In an induction proof for $\sum_{k = 1}^n \frac{1}{k^2} \leq 2 - \frac{1}{n}$ (for $n \geq 1$), I was required to prove the inequality $\frac{1}{(n+1)^2} + \frac{1}{n+1} < \frac{1}{n}$. This is my ...
4
votes
2answers
493 views

Proof by Induction for inequality, $\sum_{k=1}^nk^{-2}\lt2-(1/n)$ [duplicate]

Let $n$ be a positive natural number, $n\ge 2$. Then $\displaystyle\sum_{k=1}^n \frac{1}{k^2} \lt 2 - \frac{1}{n}.$ The basis step was easy but could someone give me a hint in the right direction as ...
4
votes
3answers
239 views

Induction on inequalities: $\frac1{1^2}+\frac1{2^2}+\frac1{3^2}+\ldots+\frac1{n^2}<2$ [duplicate]

I am trying to solve this inequality by induction. I just started to learn induction this week and all the inequalities we had been solved were like an equation less than another equation (e.g. $n! \...
4
votes
2answers
161 views

Proving inequality $\ 1+\frac14+\frac19+\cdots+\frac1{n^2}\le 2-\frac1n$ using induction

Question: Prove $$\ 1+\frac{1}{4}+\frac{1}{9}+\cdots+\frac{1}{n^2}\le 2-\frac{1}{n}, \text{ for all natural } n$$ My attempt: Base Case: $n=1$ is true: I.H: Suppose $1+\frac{1}{4}+\frac{1}{9}+\...
2
votes
3answers
62 views

induction clarification about the step $n+1$

Suppose i need to prove that $\frac{1}{2^2}+\frac{1}{3^2}...+\frac{1}{n^2}<1-\frac{1}{n}$ So in the step of $n+1$, the right side becomes $<1-\frac{1}{n+1}$ or is it: $<1-\frac{1}{n}-\frac{1}...