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# Sequences and Series (20 Problems): Partial Sums, Convergence and Divergence

Please do all problems below step by step showing me everything. Do simply as possible so I can clearly understand without rework. Adult here relearning so show all work, etc. OK, some said cannot read problems, but do not have a scanner with me know, so typed them in below. Sorry for any problems, but this shopuld clear up any confusion in reading the scanned page.

Find at least 10 partial sums of the series. Graph both the sequence of the terms and the sequence of partial sums on the same screen. Does it appear that the series is convergent or divergent? If convergent, find the sum. If divergent explain why?

3) Summation symbol n=1 to infinity of 12 /((-5)^n
5) Summation symbol n=1 to infinity of tan n
7) Summation symbol n=1 to infinity of ((1/(n^1.5) - (1/(n+1)^1.5)
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9) Let a(sub n) = 2n / 3n+1
a) Find whether {a sub n} is convergent .
b) Find whether {Summation n=1 to infinity of asub n} convergent.
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Find whether the series is convergent or divergent. If convergent find the sum.

11) 3+2+(4/3) + (8/9) + ...

13) -2 + (5/2) - (25/8) + (125/32) - ....

15) Summation n=1 to infinity 5*(2/3)^(n-1)

17) Summation n=1 to infinity [(-3)^(n-1) / 4^n]

19) Summation n= 0 to infinity [ pi^n / 3^n+1]

21) Summation n=1 to infinity n / (n+5)

23) Summation n=2 to infinity 2 / (n^2 - 1)

25 summation n=2 to infinity n^2/ (n^2-1)

27) summation n=1 to infinity (3^n + 2^n) / 6^n
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Express the number as a ratio of integers.

35) 0.2 = 0.2222....

37) 3.417 = 3.1417417417......

39) 0.123456

#### Solution Preview

Please refer to the attached file. Cheers.

3)

n an Sn
1 -2.4 -2.4
2 0.48 -1.92
3 -0.096 -2.016
4 0.0192 -1.9968
5 -0.00384 -2.00064
6 0.000768 -1.99987
7 -0.00015 -2.00003
8 3.07E-05 -1.99999
9 -6.1E-06 -2
10 1.23E-06 -2

Because the series get closer and closer to -2 and the sequence approaches 0 as n gets bigger, the series converges.

5)

n an Sn
1 1.557408 1.557408
2 -2.18504 -0.62763
3 -0.14255 -0.77018
4 1.157821 0.387643
5 -3.38052 -2.99287
6 -0.29101 -3.28388
7 0.871448 -2.41243
8 -6.79971 -9.21214
9 -0.45232 -9.66446
10 0.648361 -9.0161

Because an does not ...

#### Solution Summary

Partial Sums, Convergence and Divergence of Series are investigated. The solution is detailed and well presented. The response received a rating of "5/5" from the student who originally posted the question.

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