Find the sum to .........  1*2*3/2!  +  2*4*5/3! + 3*6*7/4! + ........

pls help me....

Asked by deepanshi bansal | 1st Nov, 2014, 11:12: PM

Expert Answer:

fraction numerator 1 cross times 2 cross times 3 over denominator 2 factorial end fraction plus fraction numerator 2 cross times 4 cross times 5 over denominator 3 factorial end fraction plus fraction numerator 3 cross times 6 cross times 7 over denominator 4 factorial end fraction plus.................. space W e space k n o w comma space e equals 1 plus 1 plus fraction numerator 1 over denominator 2 factorial end fraction plus fraction numerator 1 over denominator 3 factorial end fraction plus....... equals fraction numerator 1 over denominator n factorial end fraction T h e space n to the power of t h space end exponent t e r m space o f space t h e space s e r i e s space i s t subscript n equals fraction numerator n left parenthesis 2 n right parenthesis left parenthesis 2 n plus 1 right parenthesis over denominator left parenthesis n plus 1 right parenthesis factorial end fraction equals fraction numerator 2 n squared left parenthesis n plus n plus 1 right parenthesis over denominator left parenthesis n plus 1 right parenthesis factorial end fraction equals fraction numerator 2 n cubed over denominator left parenthesis n plus 1 right parenthesis factorial end fraction plus fraction numerator 2 n squared left parenthesis n plus 1 right parenthesis over denominator left parenthesis n plus 1 right parenthesis factorial end fraction equals fraction numerator 2 n cubed plus 2 minus 2 over denominator left parenthesis n plus 1 right parenthesis factorial end fraction plus fraction numerator 2 n squared left parenthesis n plus 1 right parenthesis over denominator left parenthesis n plus 1 right parenthesis factorial end fraction equals fraction numerator 2 left parenthesis n cubed plus 1 right parenthesis minus 2 over denominator left parenthesis n plus 1 right parenthesis factorial end fraction plus fraction numerator 2 n squared left parenthesis n plus 1 right parenthesis over denominator left parenthesis n plus 1 right parenthesis factorial end fraction equals fraction numerator 2 left parenthesis n cubed plus 1 right parenthesis over denominator left parenthesis n plus 1 right parenthesis factorial end fraction minus fraction numerator 2 over denominator left parenthesis n plus 1 right parenthesis factorial end fraction plus fraction numerator 2 n over denominator left parenthesis n minus 1 right parenthesis factorial end fraction equals fraction numerator 2 left parenthesis n plus 1 right parenthesis left parenthesis n squared minus n plus 1 right parenthesis over denominator left parenthesis n plus 1 right parenthesis factorial end fraction minus fraction numerator 2 over denominator left parenthesis n plus 1 right parenthesis factorial end fraction plus fraction numerator 2 n minus 2 plus 2 over denominator left parenthesis n minus 1 right parenthesis factorial end fraction equals fraction numerator 2 left parenthesis n squared minus n plus 1 right parenthesis over denominator n factorial end fraction minus fraction numerator 2 over denominator left parenthesis n plus 1 right parenthesis factorial end fraction plus fraction numerator 2 left parenthesis n minus 1 right parenthesis over denominator left parenthesis n minus 1 right parenthesis factorial end fraction plus fraction numerator 2 over denominator left parenthesis n minus 1 right parenthesis factorial end fraction equals fraction numerator 2 n squared over denominator n factorial end fraction minus fraction numerator 2 n over denominator n factorial end fraction plus fraction numerator 2 over denominator n factorial end fraction minus fraction numerator 2 over denominator left parenthesis n plus 1 right parenthesis factorial end fraction plus fraction numerator 2 over denominator left parenthesis n minus 2 right parenthesis factorial end fraction plus fraction numerator 2 over denominator left parenthesis n minus 1 right parenthesis factorial end fraction equals fraction numerator 2 n minus 2 plus 2 over denominator left parenthesis n minus 1 right parenthesis factorial end fraction minus fraction numerator 2 over denominator left parenthesis n minus 1 right parenthesis factorial end fraction plus fraction numerator 2 over denominator n factorial end fraction minus fraction numerator 2 over denominator left parenthesis n plus 1 right parenthesis factorial end fraction plus fraction numerator 2 over denominator left parenthesis n minus 2 right parenthesis factorial end fraction plus fraction numerator 2 over denominator left parenthesis n minus 1 right parenthesis factorial end fraction equals fraction numerator 2 left parenthesis n minus 1 right parenthesis over denominator left parenthesis n minus 1 right parenthesis factorial end fraction plus fraction numerator 2 over denominator left parenthesis n minus 1 right parenthesis factorial end fraction minus fraction numerator 2 over denominator left parenthesis n minus 1 right parenthesis factorial end fraction plus fraction numerator 2 over denominator n factorial end fraction minus fraction numerator 2 over denominator left parenthesis n plus 1 right parenthesis factorial end fraction plus fraction numerator 2 over denominator left parenthesis n minus 2 right parenthesis factorial end fraction plus fraction numerator 2 over denominator left parenthesis n minus 1 right parenthesis factorial end fraction equals fraction numerator 2 over denominator left parenthesis n minus 2 right parenthesis factorial end fraction plus fraction numerator 2 over denominator left parenthesis n minus 1 right parenthesis factorial end fraction minus fraction numerator 2 over denominator left parenthesis n minus 1 right parenthesis factorial end fraction plus fraction numerator 2 over denominator n factorial end fraction minus fraction numerator 2 over denominator left parenthesis n plus 1 right parenthesis factorial end fraction plus fraction numerator 2 over denominator left parenthesis n minus 2 right parenthesis factorial end fraction plus fraction numerator 2 over denominator left parenthesis n minus 1 right parenthesis factorial end fraction equals fraction numerator 4 over denominator left parenthesis n minus 2 right parenthesis factorial end fraction plus fraction numerator 2 over denominator left parenthesis n minus 1 right parenthesis factorial end fraction plus fraction numerator 2 over denominator n factorial end fraction minus fraction numerator 2 over denominator left parenthesis n plus 1 right parenthesis factorial end fraction...... left parenthesis i right parenthesis sum fraction numerator 4 over denominator left parenthesis n minus 2 right parenthesis factorial end fraction equals 4 open square brackets fraction numerator 1 over denominator left parenthesis minus 1 right parenthesis factorial end fraction plus 1 plus 1 plus fraction numerator 1 over denominator 2 factorial end fraction plus....... close square brackets equals 4 e space left square bracket N e g l e c t i n g space t h e space f i r s t space t e r m space o f space t h e space s e r i e s right square bracket sum fraction numerator 2 over denominator left parenthesis n minus 1 right parenthesis factorial end fraction equals 2 open square brackets 1 plus 1 plus fraction numerator 1 over denominator 2 factorial end fraction plus....... close square brackets equals 2 e sum fraction numerator 2 over denominator n factorial end fraction equals 2 open square brackets 1 plus fraction numerator 1 over denominator 2 factorial end fraction plus....... close square brackets equals 2 left parenthesis e plus 1 right parenthesis sum fraction numerator 2 over denominator left parenthesis n plus 1 right parenthesis factorial end fraction equals 2 open square brackets fraction numerator 1 over denominator 2 factorial end fraction plus....... close square brackets equals 2 left parenthesis e plus 2 right parenthesis S u b s t i t u t i n g space i n space left parenthesis i right parenthesis comma t subscript n equals fraction numerator 4 over denominator left parenthesis n minus 2 right parenthesis factorial end fraction plus fraction numerator 2 over denominator left parenthesis n minus 1 right parenthesis factorial end fraction plus fraction numerator 2 over denominator n factorial end fraction minus fraction numerator 2 over denominator left parenthesis n plus 1 right parenthesis factorial end fraction sum t subscript n equals sum open square brackets fraction numerator 4 over denominator left parenthesis n minus 2 right parenthesis factorial end fraction plus fraction numerator 2 over denominator left parenthesis n minus 1 right parenthesis factorial end fraction plus fraction numerator 2 over denominator n factorial end fraction minus fraction numerator 2 over denominator left parenthesis n plus 1 right parenthesis factorial end fraction close square brackets S equals 4 e plus 2 e plus 2 left parenthesis e plus 1 right parenthesis plus 2 left parenthesis e plus 2 right parenthesis equals 10 e plus 6

Answered by Prasenjit Paul | 3rd Nov, 2014, 11:25: AM

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