Question

Show that the series is convergent. How many terms of the series do we need to add in order to find the sum to the indicated accuracy? ∑n=1/[infinity] n / (3/1)ⁿ (error < 0.0005)

Answer 1

To show that the series is convergent, the Ratio Test is applied and the limit of the ratio of successive terms is evaluated. If the limit is less than 1, the series is convergent. To find the number of terms needed for a specified accuracy, the remainder estimation for a geometric series is used and the inequality (n+1)/(3^(n+1)) < 0.0005 is solved for n.

Step-by-step explanation:

To demonstrate that the series ∑ n=1 ∞ n / (3^n) is convergent, we can utilize the Ratio Test. To use the Ratio Test, we examine the limit of the absolute value of the ratio of successive terms as n approaches infinity:

1. Identify the general term of the series, which is a_n = n / (3^n).
2. Compute a_(n+1), which is the subsequent term in the series: a_(n+1) = (n+1) / (3^(n+1)).
3. Find the limit of |a_(n+1)/a_n| as n approaches infinity.
4. If the limit L < 1, the series is convergent.

For the error estimation, we need to use the remainder estimate for the Ratio Test. If R_n is the remainder after n terms, and the series is geometric or similar to a geometric series, we use the first term not included (n+1)th term to estimate the error: R_n ≈ |a_(n+1)|. We want the error to be less than 0.0005, so we set |a_(n+1)| < 0.0005 and solve for n.

We need to find the smallest n such that (n+1) / (3^(n+1)) < 0.0005. This usually requires trial and error or computational methods.