Series convergence

Question

Determine all integers b for which the series [(-1)ⁿn^b]/[(n-1)!] converges.

For this question would we use the ratio test?

Answer 1

Hello Lee,

Yes! Your intuition to use the ratio test is dead on! If you see n!, (n-1)!, (2n)!, kⁿ, aⁿ, products like n^b/n!, anything resembling a power series, the ratio test is almost always the correct and easiest tool.

Step 1) let a_n=((-1)ⁿn^b)/((n-1)!)

Step 2) Compute abs(a_n+1/a_n) = (n+1)^b/n^b*(n-1)!/n!=(1+1/n)^b*1/n

Step 3) As n→∞, (1+1/n)^b→1

Step 4) lim_n→∞abs(a_n+1/a_n)=lim_n→∞1/n=0

Step 5) Since the limit is 0<1, the series coverges absolutely for any integer b. Ergo, if b is any real number, the series converges because the ratio test limit is 0.

Hope this helps!!!

-Andrew Evans