Question

\[\text{Let } a_1,a_2, . . . \text {, an be an arithmetic progression with common difference }d\]

Answer 1

\[\LARGE \color{brown}{\text{compute }\sum_{k=0}^\infty \frac{ 1 }{ a_ka_{k+1} }}\]

Answer 2

multiply divide by d
note that
a_(k+1) - a_k = d

Answer 3

more, please, not understand yet

Answer 4

\[\Large \frac{ 1 }{ a_{k+1}a_k } \times \frac{ a_{k+1}-a_k }{ a_{k+1}-a_k }=(\frac{ 1 }{ a_k }-\frac{ 1 }{ a_{k+1} })\frac{ 1 }{ a_{k+1}-a_k }\]

Answer 5

i think theres telescoping

Answer 6

leave that 1/(a_(k+1) - a_k) = 1/d

now simplify the inner bracket part by just putting values of k 1 by 1