ISI MATHS QN

[tania]

CONSIDER THE SEQUENCE:
 
                         Un=summation(r=1to n) (r/2^r)
then the limit of Un as n tends to infinity is?
1) 1
2) 2
3) e
4) 1/2

PLZZZZ EXPLAIN D ANSWER!

THANKS

[Gurpreet Pannu]

Approach 1 :- intutively (if you do not remember summation of an infinite arithmetico-geometric series)

If you expand the summation you will notice that the 1st term is 1/2 also the second term too is 1/2.
These two terms itself rules out option a and d.
Now you're left with 2 or e as your answer. the 3rd term is 3/16 and the 4th term is 1/8 (0.125) and further on the terms keep on decreasing. There is no way the sum will reach e. Hence I would go with 2.

Approach 2:

Sum = a/(1-r)+ dr/(1-r)^2

Here take 1/2 common from the sum. Hence S= 1/2(1 + 2/2 +3/2^2 + 4/2^3 +......)

now a=1 d=1 and r=1/2

you will get on subsituting S= 1/2(2+2) = 2

[onionknight]

[tania]

Got it ! Thanks a lot!