Find the interval of convergence of the series. ∑(sinx/10)^n

check out this video

https://www.youtube.com/watch?v=01LzAU__J-0

To find the interval of convergence of the series ∑((sinx)/10)^n, we can start by applying the ratio test. The ratio test states that if the limit of the absolute value of the ratio of consecutive terms is less than 1, then the series converges.

Let's apply the ratio test to our series:

An = ((sinx)/10)^n
An+1 = ((sinx)/10)^(n+1)

Now, let's take the ratio of consecutive terms:

|An+1 / An| = |(((sinx)/10)^(n+1)) / (((sinx)/10)^n)|
= |(sinx/10)^(n+1) * 10^(n) / (sinx/10)^n|
= |(sinx/10) * 10^(n) / (sinx/10)^n|
= |10 * (sinx/10)^(n) / (sinx/10)^n|
= |10|

We can see that the ratio of consecutive terms is a constant value of 10, which is greater than 1. According to the ratio test, this means that the series diverges.

Therefore, the interval of convergence of the series ∑((sinx)/10)^n is empty, or the series does not converge for any value of x.