- #1
BraedenP
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Homework Statement
I am asked to determine whether a series converges, and if so, to provide its sum.
The problem is:
[tex]\sum_{n=1}^{\infty}(-3)^{n-1}4^{-n}[/tex]
Homework Equations
- I know that if the limit of the sequence as n->inf is finite, then the series converges at that limit.
- I also know that the sum of the given series is equal to said limit (if it exists).
The Attempt at a Solution
1. I rewrote the series as a limit and turned it into a fraction: [tex]\lim_{n\rightarrow\infty}\frac{(-3)^{n-1}}{4^n}[/tex]
2. Then as [itex]n\rightarrow\infty[/itex] the denominator approaches infinity, and the numerator begins to oscillate (for even n values, it's a large positive number, and for odd n values it's a large negative number).
3. Thus, I concluded that the limit does not exist, and thus the series does not converge.
However, this limit actually evaluates to 0, meaning that the series converges and sums up to 0. How is this? I'm quite confused...