Hello, jmytil!
Darn, I wish LaTeX was back on-line . . .
. . .∞ .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!