How do you test the series for convergence or divergence for #sum (-1)^n n / 10^n# from n=1 to infinity?

Answer 1

Series is absolutely convergent.

Apply root test, L= lim n#->##oo# #abs(a_(n+1)/a_n#
=lim n#->oo# #abs ((n+1)/10^(n+1)*10^n /n#
=lim n#->oo# #abs(1+1/n)*1/10#
=#1/10#
L= #1/10# < 1

Hence the series would converge

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Answer 2
To test the series for convergence or divergence for \( \sum_{n=1}^{\infty} (-1)^n \frac{n}{10^n} \), we can use the ratio test. The ratio test states that if \( \lim_{n \to \infty} \left| \frac{a_{n+1}}{a_n} \right| < 1 \), then the series converges absolutely. If \( \lim_{n \to \infty} \left| \frac{a_{n+1}}{a_n} \right| > 1 \) or if the limit is infinity, then the series diverges. If the limit equals 1, the test is inconclusive. Let's apply the ratio test to the given series: \( a_n = (-1)^n \frac{n}{10^n} \) \( a_{n+1} = (-1)^{n+1} \frac{n+1}{10^{n+1}} \) \( \lim_{n \to \infty} \left| \frac{a_{n+1}}{a_n} \right| = \lim_{n \to \infty} \left| \frac{(-1)^{n+1} \frac{n+1}{10^{n+1}}}{(-1)^n \frac{n}{10^n}} \right| \) \( = \lim_{n \to \infty} \left| \frac{(n+1)}{10(n)} \right| \) \( = \lim_{n \to \infty} \left| \frac{n+1}{10n} \right| \) \( = \lim_{n \to \infty} \left| \frac{1+\frac{1}{n}}{10} \right| \) As \( n \) approaches infinity, \( \frac{1}{n} \) approaches 0. Therefore, the limit simplifies to: \( = \frac{1}{10} \) Since \( \frac{1}{10} < 1 \), according to the ratio test, the series converges absolutely.
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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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