- #1
Lo.Lee.Ta.
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1. Find the sum of the convergent series:
∞
Ʃ 2/[(4n-3)(4n+1)]
n=12. Hm... Okay, so I started with the nth term test, and the denominator gets huge very fast. So I'm pretty sure it goes to zero.
So that tells us nothing other than that it does not FOR SURE diverge.
Since it has no n in the exponent or !, I'm not doing the ratio/root test.
Since it does have a log in it, I should do the p-series comparison test.This is where I'm having trouble.
I rewrote the problem as:
∞
Ʃ 2/(4n2 + 8n -3) →L'Hospital's Rule→ 0/(8n + 8)
n=1
____________________________________________________________________________
I was advised that by doing L'Hospital's rule, I could figure out which p-series to compare it to.
For instance, if I had:
∞
Ʃ ln(n/n+2)
n=1
I could do L'Hospital's Rule and say that it's: [(1/n) - 1/(9n+1)]/(1/n2)
I get the 1/n2 in the denominator because there are 2 n's in the numerator.
Therefore, I have to think about what would give me the 1/n2 value when L'Hospital's rule is used.
Then I know that -1/n would give me 1/n2 when L'Hospital's rule is used, and that is also the p-series that I compare to.
Since the exponent of the n is 1, it is a divergent series.
____________________________________________________________________________
The problem is that in my problem, when using L'Hospital's Rule, I get 0/(8n + 8).
With the zero on top, I don't know what to compare it to! If it was instead a 1, I would think to compare it to 1/n...
...but the antiderivative of 1/n would be ln|n|, which is not a p-series! So I think in this case another test is required! :/
But I don't even think I can use 1/n.
Would I compare it to 0? And what would the p-series then be?
#=_= UGH. I don't really know, as you can see! :/
Thank you so much for helping! :)
∞
Ʃ 2/[(4n-3)(4n+1)]
n=12. Hm... Okay, so I started with the nth term test, and the denominator gets huge very fast. So I'm pretty sure it goes to zero.
So that tells us nothing other than that it does not FOR SURE diverge.
Since it has no n in the exponent or !, I'm not doing the ratio/root test.
Since it does have a log in it, I should do the p-series comparison test.This is where I'm having trouble.
I rewrote the problem as:
∞
Ʃ 2/(4n2 + 8n -3) →L'Hospital's Rule→ 0/(8n + 8)
n=1
____________________________________________________________________________
I was advised that by doing L'Hospital's rule, I could figure out which p-series to compare it to.
For instance, if I had:
∞
Ʃ ln(n/n+2)
n=1
I could do L'Hospital's Rule and say that it's: [(1/n) - 1/(9n+1)]/(1/n2)
I get the 1/n2 in the denominator because there are 2 n's in the numerator.
Therefore, I have to think about what would give me the 1/n2 value when L'Hospital's rule is used.
Then I know that -1/n would give me 1/n2 when L'Hospital's rule is used, and that is also the p-series that I compare to.
Since the exponent of the n is 1, it is a divergent series.
____________________________________________________________________________
The problem is that in my problem, when using L'Hospital's Rule, I get 0/(8n + 8).
With the zero on top, I don't know what to compare it to! If it was instead a 1, I would think to compare it to 1/n...
...but the antiderivative of 1/n would be ln|n|, which is not a p-series! So I think in this case another test is required! :/
But I don't even think I can use 1/n.
Would I compare it to 0? And what would the p-series then be?
#=_= UGH. I don't really know, as you can see! :/
Thank you so much for helping! :)
Last edited: