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Determine whether the series k=02k+25k-1converges or diverges. Give the sum of the convergent series.

Short Answer

Expert verified

The series k=02k+25k-1 converges to 1003.

Step by step solution

01

Step 1. Given information.

Given a series k=02k+25k-1.

02

Step 2. Find if the series converges or not.

The seriesk=02k+25k-1can be expressed as k=02025k.

The series k=02025kis in the standard form k=0crkfor a geometric series with c=20and r=25.

The geometric series converges if and only if r<1.

Since r=25, it follows that the series role="math" localid="1648973658696" k=02k+25k-1converges.

03

Step 3. Find the value to which the series converges.

If the geometric series k=0crkconverges, it converges to c1-r.

So, the series k=02k+25k-1converges to 201-25, that is 1003.

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