Infinite series

• Oct 1st 2006, 07:02 AM
jmytil
Infinite series
Any help for the convergence or not of the infinite series below?

a. Sum[k^k/k!, {k,1,inf}]
b. Sum[k!/3^k, {k,1,inf}]

Thanks a lot!
• Oct 1st 2006, 07:28 AM
ThePerfectHacker
Quote:

Originally Posted by jmytil
a. Sum[k^k/k!, {k,1,inf}]

Use ratio test, (for suficiently large k), [it diverges]
The second one also diverges.
• Oct 1st 2006, 07:51 AM
Soroban
Hello, jmytil!

Darn, I wish LaTeX was back on-line . . .

Quote:

. . . .k^k
a) . .-----
. .k=1 .k!

Ratio Test . . .

a_(n+1) . . . . (k + 1)^{k+1} . . k! . . . . . (k + 1)^{k+1} . . 1 . . . . .(k + 1)^k
---------- . = . ------------------ . ------ . = . ------------------ . ----- . = . -----------
. .a_n . . . . . . . . (k+1)! . . . . . k^k . . . . . . k + 1 . . . . . . k^k . . . . . .k^k

. . . . . . . .= . [(k + 1)/k]^k . = . [1 + 1/k]^k

R . = . lim (1 + 1/k)^k . = . e
. . . . .k→∞

Since R > 1, the series diverges.

I'll let someone else explain part (b) . . . I need a nap!