00:01
The series where we have n going from zero to infinity of x to the n over n factorial.
00:07
Now if we take a sub n, well, equal to x to the n over n factorial, then while row here is equal to the limit as n tends to infinity of the absolute value of a sub m plus 1 over a sub n, right? the ratio test.
00:25
So this then is equal to, while, the limit as n goes to infinity, so we take a sub n plus 1, so it's x to the n plus 1 over n plus 1 factorial, and then dividing by a sub 1, a sub n is the same as multiplying by, well, 1 over a .m.
00:42
So we're multiplying here by n factorial over x sub n.
00:46
So we get here that this is then equal to the limit as n goes to infinity of just the absolute value of x over n.
00:57
So as we take the limit, let n go to infinity, well, x over infinity, the limit here is just equal to zero, and zero is less than one.
01:07
So since less is zero is less than one, and therefore by the ratio test here, the interval of convergence of f of x is going to be, well, negative infinity to infinity.
01:19
Okay, then for part b, looking at the derivative, so the derivative would be f prime of x, that's then equal to, well, the sum n going from one to infinity of, well, n times x to the n minus 1 over n factorial, right? just using the power rule here for derivatives, that's then equal to the sum n going from 1 to infinity of x to the n minus 1 over n minus 1 factorial...