series when n = 0 ? (1+sin n) / (10^n)

[paulxzt]

There's this one problem in the textbook that has the condition n = 0.

Sum of (1+sin n) / (10^n) with n = 0 to infinity.

Determine whether the series ir convergent or divergent.

How would I go about doing this? All the other problems/examples had n =1
thanks.

 

[Deleted member 4993]

Re: series when n = 0 ?

There's this one problem in the textbook that has the condition n = 0.

Sum of (1+sin n) / (10^n) with n = 0 to infinity.

Determine whether the series ir convergent or divergent.

How would I go about doing this? All the other problems/examples had n =1
thanks.

Conditions for convergence or divergence does not depend on the "first" number.

Sum of (1+sin n) / (10^n) with n = 0 to infinity. = (1+0)/(10^0) + Sum of (1+sin n) / (10^n) with n = 1 to infinity.

Sum of (1+sin n) / (10^n) with n = 0 to infinity. = 1 + Sum of (1+sin n) / (10^n) with n = 1 to infinity.

 

[pka]

There is a common saying among analysts: “Convergence of a series depends on what happens in its tail”. So Mr. Khan is absolutely correct.
By basic comparison, take note that
\(\displaystyle \L 0 \le \frac{{1 + \sin (n)}}{{10^n }} \le \frac{2}{{10^n }}\).

See that it converges.