- #1
Saladsamurai
- 3,020
- 7
[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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