A community for students.
Here's the question you clicked on:
 0 viewing
Loser66
 one year ago
Use induction to prove
\(\sum_{k=1}^n \dfrac{1}{k^2}\leq 2\dfrac{1}{n}\) for all positive integers n.
Please, help
Loser66
 one year ago
Use induction to prove \(\sum_{k=1}^n \dfrac{1}{k^2}\leq 2\dfrac{1}{n}\) for all positive integers n. Please, help

This Question is Closed

Loser66
 one year ago
Best ResponseYou've already chosen the best response.1HS: \(\sum_{k=1}^n \dfrac{1}{k^2}\leq 2  \dfrac{1}{n}\) Induction step \(\sum_{k=1}^{n+1} \dfrac{1}{k^2}= \sum_{k=1}^n \dfrac{1}{k^2}+\dfrac{1}{(n+1)^2}\leq 2\dfrac{1}{(n+1)^2}\)

Loser66
 one year ago
Best ResponseYou've already chosen the best response.1oh, the last one is not ^2 \(\sum_{k=1}^{n+1} \dfrac{1}{k^2}= \sum_{k=1}^n \dfrac{1}{k^2}+\dfrac{1}{(n+1)^2}\leq 2\dfrac{1}{(n+1)}\)

Loser66
 one year ago
Best ResponseYou've already chosen the best response.1This is my attempt: the first part of the sum is \(\leq 2  (1/n)\) , hence the LHS is \(\leq 2 (1/n) + (1/n+1)^2\)

Loser66
 one year ago
Best ResponseYou've already chosen the best response.1So, we need prove \(2\dfrac{1}{n} +\dfrac{1}{(n+1)^2}\leq 2 \dfrac{1}{n+1}\)

ganeshie8
 one year ago
Best ResponseYou've already chosen the best response.1clear the denominators and simplify

ganeshie8
 one year ago
Best ResponseYou've already chosen the best response.1it simplifies nicely and you end up with \(1\ge 0\)

Loser66
 one year ago
Best ResponseYou've already chosen the best response.1Yes, I did, but those are denominators, hence when I take reciprocal, the signs switch around.

Loser66
 one year ago
Best ResponseYou've already chosen the best response.1oh, let me try again. :)

ganeshie8
 one year ago
Best ResponseYou've already chosen the best response.1\(2\dfrac{1}{n} +\dfrac{1}{(n+1)^2}\leq 2 \dfrac{1}{n+1}\) \(\iff\) \(\dfrac{1}{n} \dfrac{1}{(n+1)^2}\ge \dfrac{1}{n+1}\) \(\iff\) \((n+1)^2n\ge n(n+1)\)

Loser66
 one year ago
Best ResponseYou've already chosen the best response.1I got it. Thank you so much
Ask your own question
Sign UpFind more explanations on OpenStudy
Your question is ready. Sign up for free to start getting answers.
spraguer
(Moderator)
5
→ View Detailed Profile
is replying to Can someone tell me what button the professor is hitting...
23
 Teamwork 19 Teammate
 Problem Solving 19 Hero
 Engagement 19 Mad Hatter
 You have blocked this person.
 ✔ You're a fan Checking fan status...
Thanks for being so helpful in mathematics. If you are getting quality help, make sure you spread the word about OpenStudy.