Challenging Calculus 2 question

[tame crane]

Guys can u please help me to solve this question

[distant comet]

The sum of 1/(n)(n+1) is telescoping. I haven't done the calculation, but I bet this can be done by swapping the order of the sums.

In that way, it's like many double integrals that are solved by swapping the order of integration.

[ruby lily]

I reduced it to

[sudden veldt]

animeonfire

[ruby lily]

You might be able to work out a solution from here.

[ruby lily]

If you plug the sum into desmos, you get somewhere a little over this

[sudden veldt]

animeonfire

[tame crane]

Can you please explain how you did it

[tame crane]

Thank you!

[ruby lily]

Gimme a sec, I'm gonna copy my notes over here

[sudden veldt]

animeonfire

[ruby lily]

This is what I did

And like I said, from this point on I let a calculator (in my case desmos) handle the rest.

[tame crane]

sorry for bothering you but why we write Qn as 1/2n-1 isn't it equal to the sum of 1/2n-1 from n=1 to n=infinty

[ruby lily]

Dang, yeah, you have a point.

Could you ping me if you find the solution? I'll look in here again tomorrow, but I have to get some sleep now.

[tame crane]

sure if i find ... i am trying to find for like 3 days ;D good night!

[undone mauve]

If that can help, now you are down to calculate the sum of the 1/(2k-1)k since the second terme underlined in red tends to zero

,rccw

[distant comet]

Yeah, I'm getting the same thing. I hoped that 1/(2k-1)(k) would telescope, but I'm not seeing it

[tame crane]

thank you!

[tame crane]

Thank you!

[tame crane]

can it be like that?

[distant comet]

Well done! I wouldn't have thought that factoring the 2 out of the denominator would be that useful, but it certainly is.

[undone mauve]

Wow good job, that was really smart

[tame crane]

@distant comet @undone mauve thank you, guys! 🙂

.solved