Telescoping series: I need to solve the sum from k=4 to infinity of (9/(k-1)^2) - (9/(k+2)^2)

M

MARTY122

New member
Joined
Aug 31, 2020
Messages
3
I need to solve the sum from k=4 to infinity of (9/(k-1)^2) - (9/(k+2)^2).

I cannot seem to find the partial fraction decomposition because the variables I use, A, B, C and D all have 3 factors and when setting k= either 1 or -1, all of them cancel out and I can't end up with any numbers for coefficients. Then I tried just solving w/o partial fraction decomp. but a sub 1 for example, gave me an undefined term. Does this just diverge? I'm utterly confused on what to do with this problem, and I can't find any examples like it. Please help.
 
MARTY122 said:
I need to solve the sum from k=4 to infinity of (9/(k-1)^2) - (9/(k+2)^2).

I cannot seem to find the partial fraction decomposition because the variables I use, A, B, C and D all have 3 factors and when setting k= either 1 or -1, all of them cancel out and I can't end up with any numbers for coefficients. Then I tried just solving w/o partial fraction decomp. but a sub 1 for example, gave me an undefined term. Does this just diverge? I'm utterly confused on what to do with this problem, and I can't find any examples like it. Please help.
Click to expand...
One way:
[math]S=\sum_{k=4}^\infty \dfrac{9}{(k-1)^2} - \dfrac{9}{(k+2)^2} = \sum_{k=4}^\infty \dfrac{9}{(k-1)^2} - \sum_{k=4}^\infty \dfrac{9}{(k+2)^2}[/math]
Let [imath]k-1=m[/imath] and [imath]k+2=n[/imath], then

[math]S=\sum_{m=3}^\infty \dfrac{9}{m^2} - \sum_{n=6}^\infty \dfrac{9}{n^2}[/math]
These are well-known Basel sums.
 
Second way:

[math]S=\sum_{k=4}^\infty \dfrac{9}{(k-1)^2} - \dfrac{9}{(k+2)^2} = \sum_{k=4}^\infty \dfrac{9}{(k-1)^2} - \sum_{k=4}^\infty \dfrac{9}{(k+2)^2}[/math]
Let [imath]k-1=m[/imath] then

[math]S=\sum_{m=3}^\infty \dfrac{9}{m^2} - \sum_{m=3}^\infty \dfrac{9}{(m+3)^2}[/math]
Write out the first few terms to find the partial sum [imath]S_n[/imath], then take [imath]n \to \infty.[/imath]
 
MARTY122 said:
I need to solve the sum from k=4 to infinity of (9/(k-1)^2) - (9/(k+2)^2).

I cannot seem to find the partial fraction decomposition because the variables I use, A, B, C and D all have 3 factors and when setting k= either 1 or -1, all of them cancel out and I can't end up with any numbers for coefficients. Then I tried just solving w/o partial fraction decomp. but a sub 1 for example, gave me an undefined term. Does this just diverge? I'm utterly confused on what to do with this problem, and I can't find any examples like it. Please help.
Click to expand...
It's already in partial fraction form! And there is no [imath]a_1[/imath]; the index starts at 4!

I'd just write out a few terms (you'll want at least 4 in order to start to see how it telescopes):
[math]\left(\frac{9}{3^2}-\frac{9}{6^2}\right)+\left(\frac{9}{4^2}-\frac{9}{7^2}\right)+\dots[/math]
(That's just two terms.) Then look for what cancels.
 
You must log in or register to reply here.
Top