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The Alternating Series Test and convergence.

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Functions can often be represented by an infinite series. A series representation can help to solve differential equations, to find derivatives, or to compute integrals involving the function. Computers also use these series representations to perform calculations. For example,
sinθ = θ-θ33!+θ55!-θ77!+⋯
allows a calculator to give a decimal approximation of values of sine.

The Alternating Series Test shows when an alternating series converges. Some frequently used functions have an alternating series representation. When such a series converges, that series can be used to obtain an approximation of its corresponding function.

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Biologists use exponential and logarithmic functions in growth models for populations, such as populations of bacteria. Values of such functions can be approximated with their corresponding alternating series.
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Physicists that work with optics use formulas involving sine and cosine. These formulas are simplified with the functions’ corresponding alternating series.
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Statisticians use probabilities involving bell curves and normal distributions that be approximated with alternating series for the exponential function.

The Alternating Series Test
∑ n = 1 ∞ ( − 1 ) n + 1 n
∑ n = 0 ∞ ( − 9 ) n 10 n - 2
∑ n = 0 ∞ ( − 1 ) n
Partial Sum S 1
Partial Sum
Partial Sum
Partial Sum Index Slider
Partial Sum Index Slider
Partial Sum Index Slider

INSTRUCTIONS

Click the right arrow to increase the number of terms used in the partial sum by 1.
You may also use the "Partial Sum Index Slider" or insert your own value into the box between the red and green arrows to increase/decrease the number of terms.

PRACTICE QUESTIONS

1 2 3 4
1. To apply the alternating series test, the series must be alternating. Decide if this is the case for the current series and how the fact is visualized.
2. To apply the alternating series test, the magnitude of the terms must be nonincreasing. Decide if this is the case for the current series and how the fact is visualized.
3. To apply the alternating series test, the terms must approach 0 as n approaches infinity. Is this the case for the current series?
4. Watch the display and values of Sn as you increase n to 25. If the series converges to some number L , what can we conclude about L ? Check all that apply, or select the final option if the series diverges.

INSTRUCTIONS

Click the right arrow to increase the number of terms used in the partial sum by 1.
You may also use the "Partial Sum Index Slider" or insert your own value into the box between the red and green arrows to increase/decrease the number of terms.

PRACTICE QUESTIONS

1 2 3 4
1. To apply the alternating series test, the series must be alternating. Decide if this is the case for the current series and how the fact is visualized.
2. To apply the alternating series test, the magnitude of the terms must be nonincreasing. Decide if this is the case for the current series and how the fact is visualized.
3. To apply the alternating series test, the terms must approach 0 as n approaches infinity. Is this the case for the current series?
4. Watch the display and values of Sn as you increase n to 20 . If the series converges to some number L , what can we conclude about L ? Check all that apply, or select the final option if the series diverges.

INSTRUCTIONS

Click the right arrow to increase the number of terms used in the partial sum by 1.
You may also use the "Partial Sum Index Slider" or insert your own value into the box between the red and green arrows to increase/decrease the number of terms.

PRACTICE QUESTIONS

1 2 3 4
1. To apply the alternating series test, the series must be alternating. Decide if this is the case for the current series and how the fact is visualized.
2. To apply the alternating series test, the magnitude of the terms must be nonincreasing. Decide if this is the case for the current series and how the fact is visualized.
3. To apply the alternating series test, the terms must approach 0 as n approaches infinity. Is this the case for the current series?
4. Watch the display and values of Sn as you increase n to 15 . If the series converges to some number L , what can we conclude about L ? Check all that apply, or select the final option if the series diverges.
Result

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