Question

Determine whether the series

Summation from n equals 0 to infinity e Superscript negative 5 n∑n=0∞e^−5n

converges or diverges. If it​ converges, find its sum.

Select the correct choice below​ and, if​ necessary, fill in the answer box within your choice.

A.The series converges because

ModifyingBelow lim With n right arrow infinitylimn→∞

e Superscript negative 5 ne−5nequals=0.

The sum of the series is

nothing.

​(Type an exact​ answer.)

B.The series diverges because it is a geometric series with

StartAbsoluteValue r EndAbsoluteValuergreater than or equals≥1.

C.The series diverges because

ModifyingBelow lim With n right arrow infinitylimn→∞e Superscript negative 5 ne−5nnot equals≠0

or fails to exist.

D.The series converges because it is a geometric series with

StartAbsoluteValue r EndAbsoluteValuerless than<1.

The sum of the series is

nothing.

​(Type an exact​ answer.)

E.The series converges because

ModifyingBelow lim With k right arrow infinitylimk→∞Summation from n equals 0 to k∑n=0k e Superscript negative 5 ne−5n

fails to exist.

Answer 1

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