- #1
MillerGenuine
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Homework Statement
[tex]
\sum_{n=1}^\infty(-1)^n \frac{n^n}{n!}
[/tex]
Homework Equations
I can not find my limit as n approaches infinity. I know that the answer is infinity but I am not sure how to get it.
MillerGenuine said:Seems to me that it would become [tex] \frac{\infty}{infty}=1 [/tex]
MillerGenuine said:I just based it on the fact that i have an "n" in the numerator and an "n" in the denominator so plugging in infinity for both will give me 1. Which is clearly incorrect, but i just have no idea how to approach this problem.
Don't plug in infinity like that. Your thinking is dangerous :tongue2:MillerGenuine said:Im sorry but its just not clicking for me. I see what you have shown and all i keep seeing in my head is infinity^infinity/infinity! which seems to be infinity/infinity=1
my apoogies if this sounds stupid but bare with me
[tex] \frac{\infty}{\infty} [/tex] is not a number. This is one of several indeterminate forms, others of which include 0/0, [tex] \infty - \infty, and [/tex], [tex] 1^{\infty} [/tex]MillerGenuine said:Seems to me that it would become [tex] \frac{\infty}{\infty}=1 [/tex]
I think i may take your advice on going to bed & take a fresh look at it tomorrow because i just can not seem to understand this. I am sure if either of you were to explain in person It would be much easier to understand. I think the main problem is my lack of understanding the concepts of this class. my prof only teaches mechanics of problems so i struggle with conceptual calculusRead what I posted carefully maybe something would click or go to bed and attempt the problem tomorrow
MillerGenuine said:I think i may take your advice on going to bed & take a fresh look at it tomorrow because i just can not seem to understand this. I am sure if either of you were to explain in person It would be much easier to understand. I think the main problem is my lack of understanding the concepts of this class. my prof only teaches mechanics of problems so i struggle with conceptual calculus
An alternating series is a series in which the signs of the terms alternate between positive and negative. For example, an alternating series could be written as 1 - 2 + 3 - 4 + 5 - 6 + ...
The formula for the nth term of an alternating series is (-1)^(n+1) * a_n, where a_n is the nth term of the series. This formula takes into account the alternating signs of the terms.
The Alternating Series Test is a method used to determine whether an alternating series converges or diverges. It states that if the absolute value of the terms of the series decrease as n increases and the limit of the terms approaches 0, then the series converges.
The Alternating Series Remainder is a measure of the error in approximating the sum of an alternating series. It is given by the absolute value of the (n+1)th term, where n is the number of terms used to approximate the sum. This remainder must be smaller than the desired error in order for the approximation to be accurate.
To use the Alternating Series Test, you must first check if the absolute value of the terms of the series decrease as n increases. Then, take the limit of the terms as n approaches infinity. If the limit is 0, the series converges. If the limit is not 0, the series diverges. If the limit is inconclusive, further tests may be needed to determine convergence.