# Sequences and series: exercise sheet 5

Exercise.

For each of the following, decide (with justification) which of the following series converge.

(a) =1 +1 3-2

(b) =1 -1 2-2

(c) =1 2+-1 3

(d) =1 2+-1 4-3 2+1

(e) =1 (+1)2- 2 3- 2+17

(f) =2 ( 2+-1) 12 ( 5-2) 12

(g) =2 ( 2+-1) 12 ( 3-2) 12

(h) =1 2 (Use the ratio test.)

(i) =1 (You may find it helpful to recall that (1+1) =2.71828... as .)

(j) =1 2 +3 4 +5

Exercise.

(a) Prove that =1 diverges for all 0 . (Use the null sequence test.)

(b) Prove that =1 2+1 diverges for all 1 and converges for 0 1 . (Use the ratio test.)

(c) Prove that =1 converges for all 0 .

Exercise.

Let , 0 . Show that

(a) =1 + + converges when 0 1 and diverges when 1 .

(b) +2(+1)2 2 +3(+2)3 3 + converges when 0 and diverges when .

Exercise.

Discuss the convergence of

12 +1325 2 +135 258 3 +1357 25811 4 +

## Solution to selected parts of these exercises

### Exercise 5.1(a)

The series has positive terms for 2 and

+1 3-2 2 3/2 =4 2

for 2 , so =2 +1 3-2 converges by comparison test with =2 1 2 hence =1 +1 3-2 also converges. (OR: you could use the limit comparison test here.)

### Exercise 5.1(b)

The series has positive terms for 2 and

-1 2-2 /22 2 =14 1

for 2 , so =2 -1 2-2 diverges by comparison test with =2 1 hence =1 -1 2-2 also diverges. (OR: you could use the limit comparison test here.)

### Exercise 5.3(a)

For 1 0 , the terms in the series =1 + + are all positive. Let = + + . We apply the ratio test.

+1 = ++1 ++1 +1 + + - =( +1+1 )( +1+1 ) ( +1)( +1)

which converges to by continuity of the arithmetic operations and 1 0. Therefore by the ratio test, the series converges if 1 and diverges if 1 . If =1 the series is =1 + + and + + = /+1 /+1 1 so the series diverges by the null sequence test.