Question #13952

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Answer 1 · 2017-12-07T18:02:23

Diverges.

Rearranging a little:

#(1/3)sum_(k=0)^oo(5^k)#

This is a geometric series with #r=5#. Since #r>1# we know that the series diverges.

Answer 2 · 2017-12-07T18:10:28

Divergent.

This is a geometric series with common ratio of #5#.

The sum of a geometric series is given by:

#a((1-r^n)/(1-r))#

Where #a# is the first term, #r# is the common ratio and #n# is the nth term.

If #|r|<1#

Then the sum to infinity is:

#a/(1-r)#

In this example:

The common ratio #r# is #5#:

#|5|>1#

So the series diverges.