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CBSE Class 11-science Answered

Q1. Sum of the series  1/1.2.3+5/3.4.5+9/5.6.7+.......till infinity Q2. Find value of  Log(1+1/n)^n 
Asked by Vidushi412 | 30 Nov, 2018, 09:01: AM
answered-by-expert Expert Answer
begin mathsize 12px style L e t space S space equals space fraction numerator 1 over denominator 1 cross times 2 cross times 3 end fraction plus fraction numerator 5 over denominator 3 cross times 4 cross times 5 end fraction plus fraction numerator 9 over denominator 5 cross times 6 cross times 7 end fraction.................. t o space infinity space space space space space space space space........... left parenthesis 1 right parenthesis

n to the power of t h end exponent space t e r m space o f space s e r i e s space t subscript n space equals space fraction numerator 1 plus left parenthesis n minus 1 right parenthesis 4 over denominator open parentheses 2 n minus 1 close parentheses open parentheses 2 n close parentheses open parentheses 2 n plus 1 close parentheses end fraction space equals space fraction numerator 4 n minus 3 over denominator open parentheses 2 n minus 1 close parentheses open parentheses 2 n close parentheses open parentheses 2 n plus 1 close parentheses end fraction
l e t space u s space w r i t e space t subscript n space a s space p a r t i a l space f r a c t i o n comma space space space t subscript n space equals space fraction numerator 4 n minus 3 over denominator open parentheses 2 n minus 1 close parentheses open parentheses 2 n close parentheses open parentheses 2 n plus 1 close parentheses end fraction space equals space fraction numerator A over denominator open parentheses 2 n minus 1 close parentheses end fraction plus fraction numerator B over denominator 2 n end fraction plus fraction numerator C over denominator open parentheses 2 n plus 1 close parentheses end fraction space........ left parenthesis 2 right parenthesis
B y space c o m p a r i n g space b o t h space t h e space s i d e s space o f space e q n. left parenthesis 2 right parenthesis comma space w e space g e t space A equals negative left parenthesis 1 divided by 2 right parenthesis comma space B equals 3 space a n d space C equals negative 5 divided by 2
h e n c e space t subscript n equals space fraction numerator left parenthesis negative 1 divided by 2 right parenthesis over denominator open parentheses 2 n minus 1 close parentheses end fraction minus fraction numerator 3 over denominator 2 n end fraction plus fraction numerator left parenthesis negative 5 divided by 2 right parenthesis over denominator open parentheses 2 n plus 1 close parentheses end fraction space........ left parenthesis 2 right parenthesis

T h e n space s u m space o f space s e r i e s comma space S space equals space open parentheses fraction numerator negative 1 over denominator 2 end fraction close parentheses open curly brackets 1 over 1 plus 1 third plus 1 fifth....... close curly brackets plus 3 open curly brackets 1 half plus 1 fourth plus 1 over 6 plus....... close curly brackets space minus open parentheses 5 over 2 close parentheses open curly brackets 1 third plus 1 fifth plus 1 over 7..... close curly brackets

i n space t h e space s e r i e s space w r i t t e n space i n space t h i r d space left curly bracket space right curly bracket comma space w e space a d d space plus 1 space a n d space s u b t r a c t space minus 1 space s o space t h a t space w e space c a n space c o m b i n e space t h e space f i r s t space left curly bracket space right curly bracket space a n d space space t h i r d space left curly bracket space right curly bracket
S space equals space space open parentheses fraction numerator negative 1 over denominator 2 end fraction close parentheses open curly brackets 1 over 1 plus 1 third plus 1 fifth....... close curly brackets plus 3 open curly brackets 1 half plus 1 fourth plus 1 over 6 plus....... close curly brackets space minus open parentheses 5 over 2 close parentheses open curly brackets 1 over 1 plus 1 third plus 1 fifth plus 1 over 7..... close curly brackets space plus open parentheses 5 over 2 close parentheses
S space equals open parentheses 5 over 2 close parentheses space minus space 3 open curly brackets 1 minus 1 half plus 1 third minus 1 fourth space........... close curly brackets space space equals open parentheses 5 over 2 close parentheses minus space 3 space ln space 2 space
end style
Send only one question per posting. So only one question is answered
Answered by Thiyagarajan K | 30 Nov, 2018, 11:52: AM

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