Calculus-Sec-9-1-Solution

9.1 Sequences and Series                  by SGLee - HSKim -SWSun

1-3. Find a formula for the general term  of the sequence, assuming that the pattern of the first few terms continues.

1.

.

Therefore, .

2.

.

Therefore, .

3.

Therefore, .

4-15. Determine whether the sequence converges of diverges. If it converges, find the limit.

4.

11.

Here,  and .

Thus, .

Hence, this series is divergent.

12.

Thus,

Hence, this series is divergent.

13.

Hence, this series is divergent.

14.

By L'Hopital's Rule,

Hence, this series is convergent.

15.

Since ,

Therefore,

Hence, this series is convergent.

16. Investigate the sequence  defined by the recurrence relation  for .

In particular, show that  .

Therefore,

17. Let .

(a) Determine whether  is convergent.

(b) Determine whether  is convergent.

(a)

Therefore,  is convergent.

(b)

Therefore,  is divergent.

18-24. Determine whether the following series is convergent or divergent. Find the sum if it is convergent.

18.

Thus,  is divergent.

19.

This is a geometric series and since , this series is convergent.

And, .

20.

But,  is not convergent and  is convergent.

Thus,  is divergent.

21.

.

Therefore,  is convergent to .

22.

By L'Hopital's Rule,

Therefore,  is divergent.

23.

Since , this geometric series with  () is convergent.

And, .

24.

As

Thus,  is divergent.

25-26. Express the number as a ratio of integers.

25.