For which values of α > 0 does the infinite series ∑n=1∞ sin(nπ/4)nα + sin(nπ/4) converge?

The value of ∑ k=1∞ (-1)k+1 k(k+1)k! is:

The sum of the infinite series (13 + 47) + (132 + 13 × 47 + 4272) + (133 + 132 × 47 + 13 × 4272 + 4373) + ... up to infinite terms is equal to:

The series 1p + (1/2)p + (1·3 / 2·4)p + ... is ___________.