1/1.3+1/2.5 +......+1/n(2n+1)

Asked by kien44123986 | 3rd Mar, 2020, 08:33: AM

Expert Answer:

To find the sum of the series 1, 1/1.3, 1/2.5, 1/3.7, ... 1/n(2n+1)
fraction numerator 1 over denominator 1 times 3 end fraction plus fraction numerator 1 over denominator 2 times 5 end fraction plus..... plus fraction numerator 1 over denominator n open parentheses 2 n plus 1 close parentheses end fraction
equals sum from k equals 1 to n of fraction numerator 1 over denominator k open parentheses 2 k plus 1 close parentheses end fraction
equals sum from k equals 1 to n of fraction numerator 2 over denominator 2 k open parentheses 2 k plus 1 close parentheses end fraction
equals 1 half sum from k equals 1 to n of fraction numerator 1 over denominator 2 k open parentheses 2 k plus 1 close parentheses end fraction
equals 1 half sum from k equals 1 to n of open square brackets fraction numerator 1 over denominator 2 k end fraction minus fraction numerator 1 over denominator 2 k plus 1 end fraction close square brackets
equals 1 half open square brackets 1 half sum from k equals 1 to n of 1 over k minus sum from k equals 1 to n of fraction numerator 1 over denominator 2 k plus 1 end fraction close square brackets
 

Answered by Renu Varma | 11th Mar, 2020, 12:12: PM