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Nilanjan Bhowmick AIR 3, CSIR NET (Earth Science)
L.sufiya Khanam posted an Question
October 08, 2020 • 00:25 am
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30 points
30 points
- IIT JAM
- Mathematics (MA)
Plz give me a detailed explanation on this 3 examples ,
(ii). V = (-1)": This sequence diverges whereas the sequence is bounded Si H - 1 V s1 Another example would be x = In n, since (iv). equ In n>co but Inn- In(n - 1) = In In 1=0. n-1 at it v). The sequence {sin(nt/2)),a diverges because the sequence is {1, 0, -1, 0, 1, 0, .), and hence it does not converge to any number.
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Deepak singh 1![best-answer]()
In first example - Vn=(-1)^n = {-1,1,-1,1...} limit point of Vn ={1,-1} (because 1 and -1 is coming infinitely times ) . Since limit not unique implies this series is not convergent. In second example - Xn = ln(n) Since ln(n) is unbounded so ,limit n tends to infinity Xn tends to infinity and thus divergent . On the other hand sequence Xn= ln(n/n-1) tends to 0 and hence convergent . In third example - An = {Sin(nπ/2)} limit of An ={ 1,0,-1,0,1,0......} (because 1,-1 and 0 appearing infinitely times ) Since limit not unique . hence not convergent. if any doubt then ask.
for 1st and 3rd I know it is convergent. how are they divergent as in 1st example it's like 1,-1,1,-1 .then it towards negative as well as positive . how it is divergent?
a sequence is convergent only if sequences have unique limit point..
but in 1 and 3 example , limit point is not unique , so not convergent
feel free to ask again
1 and 3 are oscillating right??
yes.
then in material it's given example for divergent sequence
a sequence is called divergent if limit is infinite or -infinite..( written in your book also )
but here in example 1 and 3 , limit is not infinite . so it is not divergent..it is oscillating
in your material , there is mistake..
feel free to ask again