10.3 Alternating Series and Absolute Convergence 1-7. Test for convergence of the following alternating series:

1. .

Sol) By the Integral Test, is divergent.

By Alternating Series Test, the series is divergent.

2. Sol) Note that iff ,  By Alternating Series Test, the series is divergent.

3. .

Sol) It satisfies because and .

By the Alternating Series Test, the series is convergent. 4. .

http://matrix.skku.ac.kr/cal-lab/cal-10-3-4.html Sol) Note that var('n') u(n)=(-1)^(n+1)*n/(n+1) limit(abs(u(n)), n=+oo)

ind

5. .

Sol) By the Integral Test, is divergent.

By Alternating Series Test, the series is divergent.

6. .

Sol)  By the Ratio Test, this series is absolutely convergent.

Hence, is convergent.

7. .

Sol) By Alternating Series Test, the series is divergent.

8. For what values of , the series is convergent.

Sol) By Exercise 5, when , this series diverges.

Let and .

For , for all x.

And for , Thus, by the Alternating Series Test, for , this series is convergent. If , this series is

divergent.

9-14. Test whether the series is absolutely convergent, conditionally convergent, or divergent.

9. .

Sol)  Hence, the series is divergent. 10. .

http://matrix.skku.ac.kr/cal-lab/cal-10-3-10.html Sol)   Divergent by the Ratio test.

 var('n') u(n)=exp(-n)*factorial(n) limit(u(n+1)/u(n), n=+oo)

+Infinity

11. .

Sol) .

Since for all , we have .

Thus Let   is absolutely convergent.

By the Comparison Test, is absolutely convergent, and hence convergent.

12. .

Sol) By the root test,  is absolutely convergent, and hence convergent.

13. .

Sol) . We don't know that this is absolutely convergent. .

Here, for all and By the Alternating Series Test, is conditionally convergent.

14. .

Sol) Then This means we can not conclude convergence of this series using the Ratio Test.

Consider Here, for all and By the Alternating Series Test, is conditionally convergent.