Question

sigma n=1 to infinity 2n/(2n+1)!​

Answer 1

Answer:

It is equal to 1/2 + 2/4 + 3/8 + 4/16 + 5/32 + 6/64 + 7/128 + …

= (1/2 + 1/4 + 1/8 + 1/16 + 1/32 + …) + (1/4 + 1/8 + /1/16 + 1/32 + …) + (1/8 + 1/16 + 1/32 + …) + … [this step may be a little difficult. Add like terms and open the brackets to expand in full if you are confused]

= (1/2 + 1/4 + 1/8 + 1/16 + 1/32 + …) + 1/2*(1/2 + 1/4 + 1/8 + 1/16 + 1/32 + …) + 1/4*(1/2 + 1/4 + 1/8 + 1/16 + 1/32 + …) + 1/8*(1/2 + 1/4 + 1/8 + 1/16 + 1/32 + …) + …

= (1/2 + 1/4 + 1/8 + 1/16 + 1/32 + …) *(1 + 1/2 + 1/4 + 1/8 + 1/16 + 1/32 + …)

= 1 * 2

= 2.