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CBSE Class 11-science Answered

Q.1  Sum of series 1/1.3+1/2.5+1/3.7+1/4.9+.......till infinity. Q.2  Sum of series 5/1.2.3+7/3.4.5+9/5.6.7...till infinity 
Asked by Vidushi412 | 29 Nov, 2018, 09:35: 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 3 end fraction plus fraction numerator 1 over denominator 2 cross times 5 end fraction plus fraction numerator 1 over denominator 3 cross times 7 end fraction plus................ space t o space infinity space space space.................. left parenthesis 1 right parenthesis space space space space space space space space space space space space
n to the power of t h end exponent space t e r m space o f space t h e space a b o v e space s e r i e s comma space t subscript n space equals space fraction numerator 1 over denominator n open parentheses 2 n plus 1 close parentheses end fraction space equals space fraction numerator 2 over denominator 2 n end fraction minus fraction numerator 2 over denominator open parentheses 2 n plus 1 close parentheses end fraction space space equals space 2 open square brackets fraction numerator 1 over denominator 2 n end fraction space minus space fraction numerator 1 over denominator open parentheses 2 n plus 1 close parentheses end fraction close square brackets
H e n c e space t h e space s u m space o f space s e r i e s space i s space w r i t t e n space a s space

S space equals space 2 open square brackets 1 half minus 1 third plus 1 fourth minus 1 fifth plus 1 over 6 minus 1 over 7 plus.......... t o space infinity close square brackets space............. space left parenthesis 2 right parenthesis
w e space k n o w comma space space ln space 2 space equals space 1 space minus 1 half plus 1 third minus 1 fourth plus 1 fifth minus 1 over 6 plus............... t o space infinity space............. left parenthesis 3 right parenthesis
u sin g space e q n. left parenthesis 2 right parenthesis space a n d space left parenthesis 3 right parenthesis comma space w e space c a n space w r i t e comma space space space space ln space 2 space equals space 1 space minus space S over 2 space space space space o r space space S space equals space 2 space space minus space 2 ln space 2 space equals space 2 space minus space ln space 4 end style
 
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begin mathsize 12px style L e t space S space equals space fraction numerator 5 over denominator 1 cross times 2 cross times 3 end fraction plus fraction numerator 7 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 5 plus left parenthesis n minus 1 right parenthesis 2 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 2 n plus 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 2 n plus 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 2 comma space B equals negative 3 space a n d space C equals 1
h e n c e space t subscript n equals space fraction numerator 2 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 1 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 2 open curly brackets 1 over 1 plus 1 third plus 1 fifth....... close curly brackets minus 3 open curly brackets 1 half plus 1 fourth plus 1 over 6 plus....... close curly brackets space plus 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 2 open curly brackets 1 over 1 plus 1 third plus 1 fifth....... close curly brackets minus 3 open curly brackets 1 half plus 1 fourth plus 1 over 6 plus....... close curly brackets space plus open curly brackets 1 over 1 plus 1 third plus 1 fifth plus 1 over 7..... close curly brackets space minus space 1
S space equals space 3 open curly brackets 1 minus 1 half plus 1 third minus 1 fourth space........... close curly brackets space minus space 1 space equals space 3 space ln space 2 space minus 1
end style
Answered by Thiyagarajan K | 30 Nov, 2018, 05:14: AM

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