infinity of the summation n=0: ((n+2)/(10^n))*((x-5)^n)
.. my work so far. i used the ratio test
= lim (n-->infinity) | [((n+3)/(10^(n+1)))*((x-5)^(n+1))] / [((n+2)/(10^n))*((x-5)^n)] |
.. now my question is: was it ok for me to add "+1" to "n+2" to become "n+3"?
= lim (n-->infinity) | [((n+3)/(10^(n+1)))*(((x-5)^(n+1))/1)] * [((10^n)/(n+2))*(1/((x-5)^n))] |
= lim (n-->infinity) | [(((n+3)(x-5))/10)*(1/(n+2))] |
how do i finish this so that i could find the endpoints? please help. thank you.
nvmd. i think i got it.
Actually, you made a mistake in your calculations. Let's go through the steps again to find the answer to your question.
You started with the series:
∑[n=0 to ∞] ((n+2)/(10^n))*((x-5)^n)
To test for convergence, you used 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, the series converges.
Next, you evaluated the ratio of consecutive terms:
lim (n --> ∞) | [((n+3)/(10^(n+1)))*((x-5)^(n+1))] / [((n+2)/(10^n))*((x-5)^n)] |
At this point, you correctly expanded the expressions and simplified them to:
lim (n --> ∞) | [(((n+3)(x-5))/10)*(1/(n+2))] |
To find the limit, we can divide numerator and denominator by n+2:
lim (n --> ∞) | [(x-5)*((n+3)/(n+2))/10] |
Since the limit of (n+3)/(n+2) as n approaches infinity is equal to 1, we have:
lim (n --> ∞) | [(x-5)*1/10] |
| (x-5)/10 |
Therefore, for the series to converge, we need |(x-5)/10| < 1. This implies:
-(x-5)/10 < 1 and (x-5)/10 < 1
Simplifying these inequalities, we have:
-x+5 < 10 and x-5 < 10
Adding 5 to both sides of the second inequality, we get:
x < 15
Therefore, the series converges if x is between 5 and 15.
If you have any further questions or need clarification, feel free to ask!