- #1

Saladsamurai

- 3,019

- 6

[tex]\sum_{k=1}^{\infty}(\frac{1}{5^k}-\frac{1}{k(k+1)})[/tex]

Now by partial fractions and distributing the sum across all three terms I get

[tex]\sum_{k=1}^{\infty}\frac{1}{5^k}-\sum_{k=1}^{\infty}\frac{1}{k}+\sum_{k=1}^{\infty}\frac{1}{k+1}[/tex]

Then I am going with: 1st is geometric; 2nd is Harmonic; and 3rd is similar to Harmonic So, 2nd and 3rd diverge.

So the sum should equal [tex]\frac{a}{1-r}[/tex].

But this the not match the text answer, what am I doing wrong?

Thanks,

Casey

EDIT: I noticed an example in my text in which I use (1/k)-1/(k+1) to write the closed form and get a finite answer of 1.

Why is this the case if I can distribute the sigma across the terms.

Now by partial fractions and distributing the sum across all three terms I get

[tex]\sum_{k=1}^{\infty}\frac{1}{5^k}-\sum_{k=1}^{\infty}\frac{1}{k}+\sum_{k=1}^{\infty}\frac{1}{k+1}[/tex]

Then I am going with: 1st is geometric; 2nd is Harmonic; and 3rd is similar to Harmonic So, 2nd and 3rd diverge.

So the sum should equal [tex]\frac{a}{1-r}[/tex].

But this the not match the text answer, what am I doing wrong?

Thanks,

Casey

EDIT: I noticed an example in my text in which I use (1/k)-1/(k+1) to write the closed form and get a finite answer of 1.

Why is this the case if I can distribute the sigma across the terms.

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