Find the interval of convergence of sumn=0^infty...

Find the interval of convergence of $$sum_{n=0}^{infty} frac{(-5)^n}{n!}(x+10)^n$$ (Use symbolic notation and fractions where needed. Give your answers as intervals in the form (*, *). Use the symbol ? for infinity, ? for combining intervals, and an appropriate type of parenthesis " (", ") " ," [" or "] " depending on whether the interval is open or closed.) x ?

Transcript

00:01 Hello everyone we are going to solve a question and this question we are given a series are summation of n from 0 to infinity here we have minus 5 raised to the power n divided by in factorial multiplied with x plus 10 raised to the power n and in this question we have to find interval of convergence interval of convergence convergence convergence now moving to the solution of this here we have a n has equals to minus 5 raised to the power n multiplied with x plus 10 raised to the power n divided by n factorial so we have the term a n plus 1 that is equals to minus 5 raised to the power n plus 1 divided by n plus 1 factorial and multiplied with x plus 10 raised to the power n plus 1 so here we find the value of limit limit and approaches to infinity of a n plus 1 divided by a n and this is equals to limit and approaches to infinity here we have minus 5 raised to the power n plus 1 x plus 10 raise to the power n plus 1 this divided by n plus 1 factorial and this whole divided by minus 5 raise to the power n divided by by n factorial and here we have x plus 10 raised to the power n so on solving this we have this is equals to limit in approaches to infinity we can write this as minus 5 raised to the power n multiplied with 5 raised to the power 1 x plus 10 raised to the power n multiplied with x plus 10 divided by n factorial by taking n common from here and this whole divided by minus 5 raised to the power n multiplied with x plus 10 raised to the power n and this divided by n factorial so on solving on canceling all the values which got cancel this is equals to limit n approaches to infinity we left with minus 5 multiplied with x plus 10 and this divided by 1 plus 1 divided by n factorial now putting the value of n as infinity so this is equals to minus 5 x plus 10 so now let this is the value of limit l if this is the limit l then series converges series converges converges if l is less than 1 so from here we have the modulus of minus 5 x plus 10 is less than 1 so from here we so when we write this without modelers, here we have minus 1 minus 5 multiplied with x plus 10.

03:06 Here this is less than 1.

03:08 So here by dividing with minus 5 we have 1 divided by 5.

03:14 Inequality sign got opposite.

03:16 Here we had x plus 10 and greater than 1 divided by minus 5...

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