For what value(s) of r is the series the summation from n equals 1 to infinity of the quotient of 1 and n raised to the quantity r plus 3 power convergent?

Question

For what value(s) of r is the series  convergent?

A	r > -2
B	r > -1
C	r ≥ -2
D	All values of r

Answer 1

A is correct

the terms are (1/n)^(r+3) with r+3 the common ratio

r+3 must be > 1 so

r+3>1

subtract 3 from both sides

r>-2

to prove the exponent must be greater than 1, use the Cauchy Condensation Test

here, the exponent is r+3 which must be greater than 1

It's a geometric series with common ratio = r+3

Answer 2

This is of the form for a_n = 1/n^p. p has to be greater than 1 for the series to be convergent.

So p = r+ 3 > 1 --> r >-2 which is answer (A)