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Asked by dineshchem108 6th December 2018, 10:43 PM
Answered by Expert
Answer:
begin mathsize 12px style t subscript r space equals space fraction numerator r over denominator 4 r to the power of 4 plus 1 end fraction space equals space fraction numerator r over denominator 4 r to the power of 4 plus 4 r squared plus 1 space minus space 4 r squared end fraction space equals space fraction numerator r over denominator open parentheses 2 r squared plus 1 close parentheses squared minus open parentheses 2 r close parentheses squared end fraction space equals space fraction numerator r over denominator open parentheses 2 r squared plus 2 r plus 1 close parentheses open parentheses 2 r squared minus 2 r plus 1 close parentheses end fraction
L e t space u s space r e s o l v e space t h e space a b o v e space f r a c t i o n a l space e x p r e s s i o n space i n t o space p a r t i a l space f r a c t i o n.

fraction numerator r over denominator open parentheses 2 r squared plus 2 r plus 1 close parentheses open parentheses 2 r squared minus 2 r plus 1 close parentheses end fraction space equals space fraction numerator A over denominator open parentheses 2 r squared plus 2 r plus 1 close parentheses end fraction space plus space fraction numerator B over denominator open parentheses 2 r squared minus 2 r plus 1 close parentheses end fraction

w e space g e t comma space space r space equals space 2 open parentheses A plus B close parentheses r squared plus 2 left parenthesis B minus A right parenthesis r plus left parenthesis A plus B right parenthesis
H e n c e space A plus B space equals 0 space semicolon space 2 left parenthesis B minus A right parenthesis space equals 1 space space
s o l v i n g space t h e space a b o v e space e q n s. comma space w e space g e t space B space equals space 1 fourth space a n d space A space equals space minus 1 fourth
H e n c e comma space space S space equals space sum from r equals 1 to n of fraction numerator r over denominator 4 r to the power of 4 plus 1 end fraction space equals space 1 fourth open square brackets sum from r equals 1 to n of fraction numerator 1 over denominator open parentheses 2 r squared minus 2 r plus 1 close parentheses end fraction space minus space space sum from r equals 1 to n of fraction numerator 1 over denominator open parentheses 2 r squared plus 2 r plus 1 close parentheses end fraction space close square brackets space............ left parenthesis 1 right parenthesis

L e t space u s space e x a m i n e space t h e space d e n o m i n a t o r space t e r m space o f space f i r s t space S u m space o f space s e r i e s

D e n o m i n a t o r space o f space r to the power of t h end exponent space t e r m space o f space f i r s t space S e r i e s space equals space open parentheses 2 r squared minus 2 r plus 1 close parentheses
D e n o m i n a t o r space o f space left parenthesis r plus 1 right parenthesis to the power of t h end exponent space t e r m space o f space f i r s t space s e r i e s space equals space 2 open parentheses r plus 1 close parentheses squared minus 2 open parentheses r plus 1 close parentheses plus 1 space equals space 2 r squared plus 2 r plus 1
H e n c e space d e n o m i n a t o r space o f space left parenthesis r plus 1 right parenthesis t h space t e r m space o f space f i r s t space s e r i e s space i s space e q u a l space t o space d e n o m i n a t o r space o f space r to the power of t h end exponent t e r m space o f space s e c o n d space s e r i e s
H e n c e space 2 to the power of n d space end exponent t e r m space o f space f i r s t space s e r i e s space w i l l space b e space c a n c e l l e d space b y space f i r s t space t e r m space o f space s e c o n d space s e r i e s
3 to the power of r d end exponent space t e r m space o f space f i r s t space s e r i e s space w i l l space b e space c a n c e l l e d space b y space 2 to the power of n d end exponent space t e r m space o f space s e c o n d space s e r i e s
4 to the power of t h space end exponent t e r m space o f space f i r s t space s e r i e s space w i l l space b e space c a n c e l l e d space b y space 3 to the power of r d end exponent space t e r m space o f space s e c o n d space s e r i e s
s o space o n
H e n c e space i n space t h e space S u m m a t i o n space g i v e n space b y space e q n. left parenthesis 1 right parenthesis comma space w e space a r e space l e f t space w i t h space o n l y space f i r s t space t e r m space left parenthesis space r equals 1 right parenthesis space o f space f i r s t space s e r i e s space

f i r s t space t e r m space space left parenthesis r space equals space 1 right parenthesis space equals space fraction numerator 1 over denominator open parentheses 2 r squared minus 2 r plus 1 close parentheses end fraction space equals 1 space space space space space comma space h e n c e space S space equals sum from r equals 1 to n of fraction numerator r over denominator 4 r to the power of 4 plus 1 end fraction space equals space 1 fourth


end style
Answered by Expert 7th December 2018, 6:06 AM
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