What if alternating series test fails

A powerful convergence theorem exists for other alternating series that meet a few conditions. Determine if the Alternating Series Test applies to each of the following series. These two series converge to their sums at different rates. To be accurate to two places after the decimal, we need terms of the first series though only 13 of the second.

To get 3 places of accuracy, we need terms of the first series though only 33 of the second. Why is it that the second series converges so much faster than the first? While there are many factors involved when studying rates of convergence, the alternating structure of an alternating series gives us a powerful tool when approximating the sum of a convergent series.

Some alternating series converge slowly. Notice how the first series converged quite quickly, where we needed only 10 terms to reach the desired accuracy, whereas the second series took over 9, terms. The notion that alternating the signs of the terms in a series can make a series converge leads us to the following definitions. Determine if the following series converge absolutely, conditionally, or diverge. Knowing that a series converges absolutely allows us to make two important statements, given in the following theorem.

The first is that absolute convergence is "stronger'' than regular convergence. One reason this is important is that our convergence tests all require that the underlying sequence of terms be positive. By taking the absolute value of the terms of a series where not all terms are positive, we are often able to apply an appropriate test and determine absolute convergence.

This, in turn, determines that the series we are given also converges. The second statement relates to rearrangements of series. When dealing with a finite set of numbers, the sum of the numbers does not depend on the order which they are added.

One may be surprised to find out that when dealing with an infinite set of numbers, the same statement does not always hold true: some infinite lists of numbers may be rearranged in different orders to achieve different sums. The theorem states that the terms of an absolutely convergent series can be rearranged in any way without affecting the sum.

In Example 8. Theorem 72 tells us the series converges which we could also determine using the Alternating Series Test. The theorem states that rearranging the terms of an absolutely convergent series does not affect its sum. This implies that perhaps the sum of a conditionally convergent series can change based on the arrangement of terms. Alternating series test question. If it fails, what does it mean? Ask Question. Asked 2 years, 7 months ago. Active 2 years, 7 months ago. Viewed 2k times.

Jwan Jwan 5, 6 6 gold badges 39 39 silver badges 76 76 bronze badges. What you've done here looks all right to me. The failure of the AST shows really nothing? Add a comment. Active Oldest Votes. Travis Willse Travis Willse Sonnhard Graubner Dr. Sonnhard Graubner 1. Sign up or log in Sign up using Google. Sign up using Facebook. Sign up using Email and Password. Post as a guest Name. Email Required, but never shown.

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