Find the sum of the following infinite series:
2/5 + 3/5*5 + 2/5*5*5 + 3/5*5*5*5.....

Asked by s_patne2002 | 15th Nov, 2015, 02:29: PM

Expert Answer:

C o n s i d e r space infinity infinity t h e space g i v e n space s e r i e s colon S equals 2 over 5 plus fraction numerator 3 over denominator 5 cross times 5 end fraction plus fraction numerator 2 over denominator 5 cross times 5 cross times 5 end fraction plus fraction numerator 3 over denominator 5 cross times 5 cross times 5 cross times 5 end fraction plus fraction numerator 2 over denominator 5 cross times 5 cross times 5 cross times 5 cross times 5 end fraction plus... space space space equals 2 over 5 plus fraction numerator 2 over denominator 5 cross times 5 cross times 5 end fraction plus fraction numerator 2 over denominator 5 cross times 5 cross times 5 cross times 5 cross times 5 end fraction plus... plus fraction numerator 3 over denominator 5 cross times 5 end fraction plus fraction numerator 3 over denominator 5 cross times 5 cross times 5 cross times 5 end fraction plus fraction numerator 3 over denominator 5 cross times 5 cross times 5 cross times 5 cross times 5 cross times 5 end fraction plus... space space space equals S subscript 1 plus S subscript 2 comma space space w h e r e space S subscript 1 equals 2 over 5 plus fraction numerator 2 over denominator 5 cross times 5 cross times 5 end fraction plus fraction numerator 2 over denominator 5 cross times 5 cross times 5 cross times 5 cross times 5 end fraction plus... a n d space S subscript 2 equals fraction numerator 3 over denominator 5 cross times 5 end fraction plus fraction numerator 3 over denominator 5 cross times 5 cross times 5 cross times 5 end fraction plus fraction numerator 3 over denominator 5 cross times 5 cross times 5 cross times 5 cross times 5 cross times 5 end fraction plus... rightwards double arrow S equals sum from k equals 1 to infinity of 2 over 5 to the power of 2 k minus 1 end exponent plus sum from k equals 1 to infinity of 3 over 5 to the power of 2 k end exponent rightwards double arrow S equals 2 sum from k equals 1 to infinity of 1 over 5 to the power of 2 k minus 1 end exponent plus 3 sum from k equals 1 to infinity of 1 over 5 to the power of 2 k end exponent.... left parenthesis 1 right parenthesis N o w space c o n s i d e r space t h e space f i r s t space s u m comma space sum from k equals 1 to infinity of 1 over 5 to the power of 2 k minus 1 end exponent colon sum from k equals 1 to infinity of 1 over 5 to the power of 2 k minus 1 end exponent space i s space a space G P comma space w h e r e comma space a equals 1 fifth space a n d space c o m m o n space r a t i o space i s space r equals 1 over 5 squared T h u s comma space S u m space o f space i n f i n i t e space t e r m s space o f space G P comma space S subscript infinity equals fraction numerator a over denominator 1 minus r end fraction space space space space space space equals fraction numerator 1 fifth over denominator 1 minus 1 over 5 squared end fraction space space space space space space equals fraction numerator 1 fifth over denominator 24 over 25 end fraction sum from k equals 1 to infinity of 1 over 5 to the power of 2 k minus 1 end exponent equals 5 over 24... left parenthesis 2 right parenthesis N o w space c o n s i d e r space t h e space s e c o n d space s u m comma space sum from k equals 1 to infinity of 1 over 5 to the power of 2 k end exponent colon sum from k equals 0 to infinity of 1 over 5 to the power of 2 k end exponent space i s space a space G P comma space w h e r e comma space a equals 1 space a n d space c o m m o n space r a t i o space i s space r equals 1 over 5 squared T h u s comma space S u m space o f space i n f i n i t e space t e r m s space o f space G P comma space S subscript infinity equals fraction numerator a over denominator 1 minus r end fraction space space space space space space equals fraction numerator 1 over denominator 1 minus 1 over 5 squared end fraction space space space space space space equals fraction numerator 1 over denominator 24 over 25 end fraction space space space space space space space equals 25 over 24... left parenthesis 3 right parenthesis sum from k equals 0 to infinity of 1 over 5 to the power of 2 k end exponent equals 1 plus sum from k equals 1 to infinity of 1 over 5 to the power of 2 k end exponent B u t comma space w e space n e e d space o n l y comma space sum from k equals 1 to infinity of 1 over 5 to the power of 2 k end exponent H e n c e comma space sum from k equals 1 to infinity of 1 over 5 to the power of 2 k end exponent equals sum from k equals 0 to infinity of 1 over 5 to the power of 2 k end exponent minus 1 rightwards double arrow sum from k equals 1 to infinity of 1 over 5 to the power of 2 k end exponent equals 25 over 24 minus 1 rightwards double arrow sum from k equals 1 to infinity of 1 over 5 to the power of 2 k end exponent equals fraction numerator 25 minus 24 over denominator 24 end fraction rightwards double arrow sum from k equals 1 to infinity of 1 over 5 to the power of 2 k end exponent equals 1 over 24 space... left parenthesis 4 right parenthesis space space space space space space S u b s t i t u t i n g space t h e space v a l u e s space o f space sum from k equals 1 to infinity of 1 over 5 to the power of 2 k minus 1 end exponent space a n d space sum from k equals 1 to infinity of 1 over 5 to the power of 2 k end exponent space f r o m space e q u a t i o n s space left parenthesis 3 right parenthesis space a n d space left parenthesis 4 right parenthesis space i n space e q u a t i o n space left parenthesis 1 right parenthesis comma space w e space h a v e comma S equals 2 cross times 5 over 24 plus 3 cross times 1 over 24 space space space equals 10 over 24 plus 3 over 24 space space space equals 13 over 24

Answered by Vimala Ramamurthy | 16th Nov, 2015, 10:38: AM