Question

Does the series (1+sin(n))/(10^n) from summation 0 to positive infinity converge or diverge?

I let an = (1+sin(n))/(10^n) and bn = 1/10^n

lim as n approaches positive infinity = an/bn = ((1+sin(n))/(10^n))/(1/(10^n))= 1+sin(n)= positive infinity.

I don't know if it's right or not but if someone could look over it that would be great.

Answer 1

I ran this simple Quick-Basic program

FOR N = 1 TO 20
TERM = (1+ SIN(N))/(10^N)
SUM = SUM + TERM
PRINT TERM, SUM
NEXT N

the terms keep getting smaller by appr a factor of 10, (obviously the denominators become 1,10,100,1000, ...)
since the sin(N) can only be a number between -1 and 1, then (1+sin(n)) will range between 0 and 2

after about 7 terms, the sum had approached a value of 1.204407 and since each successive term would be smaller than 10^-7, it would converge to that sum.