Match the letter with the formula stating the first correct reason why the serie

Question

Match the letter with the formula stating the first correct reason why the series diverges.

A. Diverges because the terms don't have limit zero
B. Divergent geometric series
C. Divergent p series
D. Integral test
E. Comparison with a divergent p series
F. Diverges by limit comparison test
G. Diverges by alternating series test

1. ?n=1? 5n+4/(?1)^n
2. ?n=1? 1/nln(n)
3. ?n=1? (n)^(-1/6)
4. ?n=1? 1/sqrt(n)?4
5. ?n=1? ln(n)/n
6. ?n=1? cos(n?)/ln(7)

For each of the series below select the letter from A to C that best applies and the letter from D to K that best applies. A possible answer is AF, for example.

A. The series is absolutely convergent.
B. The series is conditionally convergent.
C. The series diverges.

D. The alternating series test shows the series converges.
E. The series is a p-series.
F. The series is a geometric series.
G. By comparison with a p-series.
H. By comparison with a geometric series.
I. Partial sums of the series telescope.
J. The terms of the series do not have limit zero.
K. By the ratio test.

?n=2? (?8)^(n)/5^(n)+1
2. ?n=1? 7+sqrt(n)/n
3. ?n=1? (?1)^(n)/n?
4. ?n=1? 1/5^(n)+10
5. ?n=1? 1/n*sqrt(n)
6. ?n=1? (?5)^(n)/n!

Determine whether each series converges or not. For the series which converges, enter the sum of the series. For the series which diverges enter "DIV" (without quotes). These must be solved in order to get the points. If you miss a porblem but still do well you get all points. GOOD LUCK and THANKS

1. ?n=1? 8^n/7^n=   

2. ?n=0? 2^n/6^(2n+1)=   

3. ?n=1? 6/n(n+1)=

4. ?n=5? 7^n/8^n=   

5. ?n=1? 5^n/5^(n+4)=

6. ?n=1? 7^n+2^n/8^n=

Explanation / Answer

1A

2 D

3 C

4 F

5 D (though E works too)

6 A

for second part

1. CJ

2CG

3BD

4AH

5AE

6AK

part 3

1. DIV

(8/7)>1 , so it is a geometric series that diverges

2. 2^n/(6^(2n+1))=1/6*2^n/6^(2n)=1/6*2^n/32^n=1/6*1/16^n

so it is a geomteric series

with sum 1/6(1+1/16+1/16^2+...)=1/6*1/(1-1/16)=15/(6*16)=5/32

3. 1/(n(n+1))=1/n-1/(n+1) so it is a telescopic sum

with sum 6(1-1/2+1/2-1/3-...)=6

4. a geometric sum

(7/8)^5(1+7/8+(7/8)^2+...)=(7/8)^5*1/(1-7/8)=7^5/8^4

5. DIV

diverges because lim 5^n/(5^(n+4))=1 and thus is not 0

6. sum (7/8)^n+ sum (2/8)^n=7/8 (1+7/8+...)+2/8(1+2/8+...)=

7/8*1/(1-7/8)+2/8*1/(1-2/8)=7+1/3=22/3

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