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24. Which of the following is/are divergent geometric series?ΙΣ 1Σ34k36IV.ΣII.24III.2k+I22KA) I, IIB) III, IVC) I, IIID) III onlyE) None

Question
24. Which of the following is/are divergent geometric series?
ΙΣ 1Σ
34
k
36
IV.
Σ
II.
24
III.
2k+I
22K
A) I, II
B) III, IV
C) I, III
D) III only
E) None
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24. Which of the following is/are divergent geometric series? ΙΣ 1Σ 34 k 36 IV. Σ II. 24 III. 2k+I 22K A) I, II B) III, IV C) I, III D) III only E) None

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Step 1

Consider the series I.

The partial sum is,

S
k
11
3
S3
2
the partial sums tends to infinity
As n
1
diverges.
Therefore, the series
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S k 11 3 S3 2 the partial sums tends to infinity As n 1 diverges. Therefore, the series

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Step 2

Therefore, the options B, D and E are not possible.

Now consider the series II.

By series ratio test,

2ktk
1
1+
k
axtends to
As limit k
2
k
converges
Therefore, the series
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2ktk 1 1+ k axtends to As limit k 2 k converges Therefore, the series

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Step 3

Now consider the series III.

By...

ap3*2*+1
242 35
3
2
3
0. *+1 tends to
As limit k
>0
2
3k
Therefore, the series 7 diverges.
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ap3*2*+1 242 35 3 2 3 0. *+1 tends to As limit k >0 2 3k Therefore, the series 7 diverges.

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