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.

what is the answer x-5/3=10

To find the endpoints of the summation, you need to determine the conditions under which the series converges or diverges.

Using the ratio test was a good approach to analyze the convergence of the series.

The ratio test states that for a series ∑aₙ to converge, the limit of the absolute value of the ratio of consecutive terms |aₙ₊₁ / aₙ| as n approaches infinity must be less than 1.

So, let's apply the ratio test to the given series:

You correctly wrote the ratio as:
lim (n-->infinity) | [((n+3)/(10^(n+1)))*(((x-5)^(n+1))/1)] * [((10^n)/(n+2))*(1/((x-5)^n))] |

To simplify this further, let's divide the numerator and denominator by 10n:

lim (n-->infinity) | [(n+3)(x-5)/(10(n+1))] * [(1/(n+2))] |

Now, let's take the limit:

lim (n-->infinity) | (n+3)(x-5)/(10(n+1)(n+2)) |

To determine the endpoints, we need to check if the series converges for x = 5 and x ≠ 5.

For x = 5:
lim (n-->infinity) | (n+3)(5-5)/(10(n+1)(n+2)) |
= lim (n-->infinity) | 0/(10(n+1)(n+2)) |
= 0

For x ≠ 5:
lim (n-->infinity) | (n+3)(x-5)/(10(n+1)(n+2)) |
= lim (n-->infinity) | (x-5)/(10(n+1)(n+2)) |

The series converges for x ≠ 5 if and only if the limit above is less than 1. If the limit is greater than or equal to 1, the series diverges.

To determine the exact range of x values for convergence, you need to calculate the limit explicitly. However, if you're only interested in finding the endpoints, you can use the fact that if the limit is less than 1, the series converges for all x values.

So, in conclusion, the series converges for all x values except x = 5.