Sequence and series (3)

Algebra Level pending

Define T n = 1 ( n + 1 ) H n H n + 1 T_n = \dfrac{1}{(n+1)H_nH_{n+1}} , where H n = 1 + 1 2 + 1 3 + ⋯ + 1 n H_n = 1+\dfrac 12+\dfrac 13 + \cdots + \dfrac 1n denotes the n n th harmonic number .

Find T 1 + T 2 + T 3 + ⋯ T_1 + T_2 + T_3 + \cdots .

1 3 \dfrac13 1 4 \dfrac14 1 1 1 2 \dfrac12

Adhiraj Dutta by Adhiraj Dutta , 18, India

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1 solution

Chew-Seong Cheong
Apr 24, 2020

∑ n = 1 ∞ 1 ( n + 1 ) H n H n + 1 = ∑ n = 1 ∞ ( 1 H n − 1 H n + 1 ) By partial fraction decomposition = 1 H 1 = 1 \begin{aligned} \sum_{n=1}^\infty \frac 1{(n+1)H_nH_{n+1}} & = \sum_{n=1}^\infty \left(\frac 1{H_n} - \frac 1{H_{n+1}} \right) & \small \blue{\text{By partial fraction decomposition}} \\ & = \frac 1{H_1} = \boxed 1 \end{aligned}

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