[Ether]

Originally Posted by GeeTwo

How about this one:

• [as shown earlier] (k/(k+3))^2k (the terms, not the series sum) converges to e^-6 ~ .00248 ~ 1/403.
• This means that for every k sufficiently large, (k/(k+3))^2k < 1/4 (actually I believe this is true for all of them)
• As k/(k+3) is positive, (k/(k+3))^k < 1/2 for sufficiently large k.
• Therefore, for every k sufficiently large, (k/(k+3))^k2 < 1/2^k
• We know S_k=1^OO 1/2^k, converges at 1.0.
• By the comparison test, S_k=1^OO (k/(k+3))[sup]k^2[/sup converges.

Unless I'm missing something, I see no flaw there. Nice.