Asked by Tesfaye teshome
prove that 1+1/4+.........+1/n^2<2-1/n
Answers
Answered by
Steve
true for k=1?
1/1^2 < 2 - 1/1 ?
no. well, maybe
1 + 1/2^2 < 2 - 1/2 ?
5/4 < 3/2 ?
yes. So, we'll do induction for k>2
assume
∑ 1/k^2 < 2 - 1/k
now show that
∑ 1/k^2 + 1/(k+1)^2 = 2 - 1/k + 1/(k+1)^2 < 2 - 1/(k+1)^2
see what you can do with that.
I bet google can help if you still get stuck.
1/1^2 < 2 - 1/1 ?
no. well, maybe
1 + 1/2^2 < 2 - 1/2 ?
5/4 < 3/2 ?
yes. So, we'll do induction for k>2
assume
∑ 1/k^2 < 2 - 1/k
now show that
∑ 1/k^2 + 1/(k+1)^2 = 2 - 1/k + 1/(k+1)^2 < 2 - 1/(k+1)^2
see what you can do with that.
I bet google can help if you still get stuck.
There are no AI answers yet. The ability to request AI answers is coming soon!
Submit Your Answer
We prioritize human answers over AI answers.
If you are human, and you can answer this question, please submit your answer.