Could Somebody Help Me Calculate These Series? (reps)

[Masterchieffer]

1. Does the summation from k=1 to infinite of (ln k)/k^2 converge or diverge? Why?

2. Does the summation from k=2 to infinite of (k+1)/[(k^2)+2] converge or diverge? Why?

Major reps for answers and work. I have an assignment due on this tomorrow at 3, and today was the first day he even began lecturing on this stuff (if it is indeed the limit comparison stuff that I think it is).

[Masterchieffer]

Anybody?

[dat]

Evaluate the limits as k->infinity

If the function tends to infinity the summation will diverge to infinity. If the function tends to zero the summation will converge to a constant.

[kiwimac]

First one converges because ln k increases at a slower rate than k^2 and so the limit of the sequence of terms converges to 0. This implies that the series converges.

Second converges because the denominator contains a k^2 term while the numerator only has a linear (k) term. This means the sequence of terms converges and so the series converges.

Not sure if you want a formal proof or not.

Edit: The theorem actually says that if the summation converges, then the sequence of terms converge to 0. So what I said above is wrong.

[Ephedra]

Tree fiddy.

[Masterchieffer]

Originally Posted by dat

Evaluate the limits as k->infinity

If the function tends to infinity the summation will diverge to infinity. If the function tends to zero the summation will converge to a constant.

right, and the limit of the sequences both converge to zero, so the series most likely converge, but how?

[Masterchieffer]

Originally Posted by kiwimac

First one converges because ln k increases at a slower rate than k^2 and so the limit of the sequence of terms converges to 0. This implies that the series converges.

Second converges because the denominator contains a k^2 term while the numerator only has a linear (k) term. This means the sequence of terms converges and so the series converges.

Not sure if you want a formal proof or not. But there is a theorem that says that if the sequence of terms converges to a limit, then the summation (i.e. the infinite series which adds the terms up) converges. You would then use the definition of a limit to show that the sequences converge to 0.

yeah for sure, but I need a numerical proof

[Masterchieffer]

Originally Posted by kiwimac

First one converges because ln k increases at a slower rate than k^2 and so the limit of the sequence of terms converges to 0. This implies that the series converges.

Second converges because the denominator contains a k^2 term while the numerator only has a linear (k) term. This means the sequence of terms converges and so the series converges.

Not sure if you want a formal proof or not.

Edit: The theorem actually says that if the summation converges, then the sequence of terms converge to 0. So what I said above is wrong.

dud brah your phucking me

if you take the lim k--> infinite of both of those functions without the summation part you will get zero, which implies that the series converges, like you said. I'm wondering what they converge to and how.

[Masterchieffer]

bump b4 bed

[kiwimac]

Originally Posted by Masterchieffer

dud brah your phucking me

if you take the lim k--> infinite of both of those functions without the summation part you will get zero, which implies that the series converges, like you said. I'm wondering what they converge to and how.

No, the implication only goes one way. There are series for which th sequence of terms converges to zero, but the series is divergent.

"Unlike stronger convergence tests, the term test cannot prove by itself that a series converges."

http://en.wikipedia.org/wiki/N-th_term_test