Solutions to Problem Set 16

Math 211-03

10-16-2017

[Sequences]

1. Determine the limits of the geometric sequences in the table below, Your answers should be 0, 1, , or "undefined".

2. Determine whether the sequence defined for is (eventually) increasing, decreasing, or neither.

Let . Then

For , I have . Therefore, the sequence is decreasing.

3. Determine whether the sequence defined for is (eventually) increasing, decreasing, or neither.

Let . Then

For , I have , and and are positive. Therefore, for , and the sequence is (eventually) increasing.

4. Determine whether the sequence defined for is (eventually) increasing, decreasing, or neither.

Since oscillates as , the sequence is neither eventually increasing nor eventually decreasing.

5. Prove that the sequence is bounded.

If a sequence has a (finite) limit, then it must be bounded.

Since the sequence has a finite limit, it is bounded.

6. A sequence is defined recursively by

Find , assuming that the limit exists.

Let

Note that

Reason: Both of these limits give the value that the a's approach, which is L:

Take the limit on both sides of the the recursion equation:

The definition of the a's shows that all of them are positive, so their limit can't be negative. Therefore,

[Series]

7. Determine whether the series converges or diverges. If the series converges, find the exact value of its sum.

The series is geometric, with ratio . Hence, it converges, and its sum is

8. Determine whether the series converges or diverges. If the series converges, find the exact value of its sum.

The series is geometric, with ratio -0.9. Hence, it converges, and its sum is

9. Determine whether the series converges or diverges. If the series converges, find the exact value of its sum.

The series is geometric, with ratio . Hence, it diverges.

10. Determine whether the following series converges or diverges. If the series converges, find the exact value of its sum.

The series is geometric, with ratio . Hence, it converges, and its sum is

11. Determine whether the series converges or diverges. If the series converges, find the exact value of its sum.

The series is the sum of two geometric series.

The series has ratio , so it converges. Its sum is

The series has ratio , so it converges. Its sum is

Hence, the original series converges, and its sum is

12. Write the following infinite repeating decimal as a geometric series, and find the exact value of its sum:

The series is geometric with ratio . Its sum is

13. Determine whether the series converges or diverges. If the series converges, find the exact value of its sum.

Note that

The series is 7 times the harmonic series, which diverges. So the original series diverges.

14. Use telescoping to find the exact value of the sum of the series .

Hint: By partial fractions,

I have

To see what is happening, write out the first few terms of the last sum:

You can see that the terms , , and so on cancel in pairs. The only terms which don't cancel are and . So

Real stories, in distinction from those we invent, have no author. - Hannah Arendt

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