What is the interval of convergence of the infinite series (below) ... ?

Question

oo
∑ [(3x)^k]/(k!).
k=0


possible solutions....

a. (2, 4)

b. [2, 4)

c. (2, 4]

d. [-1/3, 1/3)

e. (-1/3, 1/3)

f. [-1/3, 1/3]

g. (-1, 1]

h. [-1, 1)

i. [-1, 1]

j. (-2, 2)

k. (-4, 4)

l. x = 0 only

m. x = 2 only

n. x = 4 only

o. (-oo, oo)

p. none of these

Answer 1

oo
∑ [(x - 3)^k]/(k!).
k=0
Use the ratio test.
[(x-3)^(k+1) / (k+1)!] * [k! / (x-3)^k]
= (x-3) / (k+1)
The limit as k goes to infinity of the above is zero.
This means that the radius of convergence is all real numbers for x (option o).

Answer 2

Thats right, the answer would be o since your series is just exp(3x).

Answer 3

o

its just exp(3x)