What is the limit of the sequence? a(k) = (1+2+3+...+k)/(1*2*3*...*k)

sum k=1..oo of (1+2+3+...k)/(1*2*3*...*k)

sum k=1..oo of k(k+1)/k!2

1/2 sum k=1..oo of (k+1)/(k-1)!

Setting u=k-1

1/2 sum u=0..oo of (u+2)/u!

1/2 (2 + sum u=1..oo of (u+2)/u!)

1 + 1/2 sum u=1..oo of (u+2)/u!

1 + 1/2 [(sum u=1..oo of 1/(u-1)!) + (2 sum u=1..oo of 1/u!)]

1 + 1/2 (sum u=1..oo of 1/(u-1)!) + (sum u=1..oo of 1/u!)

Setting w=u-1

1 + 1/2 (sum w=0..oo of 1/w!) + (sum u=1..oo of 1/u!)

1 + 1/2 (sum w=0..oo of 1/w!) + (sum u=0..oo of 1/u!) - 1

3/2 (sum u=0..oo of 1/u!)

3/2 e

3e/2

Since the sum series is convergent (to 3e/2), the limit must be zero.

Posted by Kurious on 2008-01-06 16:46:35