How do you use the ratio test to test the convergence of the series #∑ (n+1)/(3^n)# from n=1 to infinity?

Answer 1

By the ratio test, the series converges.

The ratio tests states that a series #sum_n^oo a_n# converges if #L<1# and diverges if #L>1#, where
#L = lim_(n->oo) |a_(n+1)/a_n|#
If the limit is #1# or doesn't exist, the test is inconclusive.
Since the general term of our series is #a_k = (k+1)/3^k#, we have:
#L = lim_(n->oo) |((n+1)/3^(n+1))/(n/3^n)|#
Note that, as #n->oo#, this limit is clearly positive therefore the absolute value is not needed.
#L = lim_(n->oo) (n+1)/3^(n+1) * 3^n/n = lim_(n->oo)3^n/(3^n*3) * (n+1)/n #
#L = lim_(n->oo) 1/3 * (1 + 1/n)#
Since #n# approaches infinity, #1/n# must approach #0#.
#L = 1/3 * (1+0) = 1/3#
#L<1#, which means that the series #color(red)("converges by the ratio test")#.
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Answer 2

To use the ratio test to test the convergence of the series ( \sum_{n=1}^{\infty} \frac{n+1}{3^n} ), you would calculate the limit:

[ \lim_{n \to \infty} \left| \frac{a_{n+1}}{a_n} \right| ]

Where ( a_n ) represents the nth term of the series. In this case, ( a_n = \frac{n+1}{3^n} ).

So, we have:

[ \lim_{n \to \infty} \left| \frac{\frac{n+2}{3^{n+1}}}{\frac{n+1}{3^n}} \right| ]

[ = \lim_{n \to \infty} \left| \frac{n+2}{3^{n+1}} \times \frac{3^n}{n+1} \right| ]

[ = \lim_{n \to \infty} \left| \frac{n+2}{3n + 3} \right| ]

Since the numerator and denominator both approach infinity as ( n ) approaches infinity, we can apply L'Hopital's Rule to evaluate the limit:

[ = \lim_{n \to \infty} \frac{1}{3} = \frac{1}{3} ]

According to the ratio test, if this limit is less than 1, the series converges absolutely. If it's greater than 1 or diverges to infinity, the series diverges. If the limit equals 1, the ratio test is inconclusive, and another test may be needed. In this case, since ( \frac{1}{3} ) is less than 1, the series ( \sum_{n=1}^{\infty} \frac{n+1}{3^n} ) 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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