$\boldsymbol{X}$ Incorrect.

The series converges by the alternating series test. How many terms give a partial sum, $S_{n}$, within 0.02 of the sum, $S$, of the series?
$$
\begin{array}{l}
\sum_{n=1}^{\infty} \frac{(-1)^{n-1}}{n} \
n \geq
\end{array}
$$
i 50
eTextbook and Media

Hint
Assistance Used

Since the series converges and is an alternating series of the form $\sum_{n=1}^{\infty}(-1)^{n-1} a_{n}$, by Theorem 9.9: Error Bounds for Alternating Series you know $\left|S-S_{n}\right|

Q: $\boldsymbol{X}$ Incorrect. The series converges by the alternating series test. How many terms give a partial sum, $S_{n}$, within 0.02 of the sum, $S$, of the series? \[ \begin{array}{l} \sum_{n=1}^{\infty} \frac{(-1)^{n-1}}{n} \\ n \geq \end{array} \] i 50 eTextbook and Media Hint Assistance Used Since the series converges and is an alternating series of the form $\sum_{n=1}^{\infty}(-1)^{n-1} a_{n}$, by Theorem 9.9: Error Bounds for Alternating Series you know $\left|S-S_{n}\right|<a_{n+1}$.

∻Solution by Steps∻ ‖ step 1 ⋮ To determine the number of terms needed to get a partial sum within 0.02 of the total sum of the series, we use the error bound for alternating series. According to Theorem 9.9, for an alternating series of the form $ \sum_{n=1}^{\infty}(-1)^{n-1} a_{n} $, the error $ \left|S - S_{n}\right| $ is less than $ a_{n+1} $. ‖ ‖ step 2 ⋮ The given series is $ \sum_{n=1}^{\infty} \frac{(-1)^{n-1}}{n} $. Here, $ a_{n} = \frac{1}{n} $. ‖ ‖ step 3 ⋮ To find the number of terms $ n $ such that the error is within 0.02, we need $ a_{n+1} 50 $, which simplifies to $ n > 49 $. ‖ ‖ step 6 ⋮ Therefore, we need at least 50 terms to ensure the partial sum is within 0.02 of the total sum. ‖ ∻Answer∻ ⚹ 50 ⚹ ∻Key Concept∻ ⚹Error Bound for Alternating Series⚹ ∻Explanation∻ ⚹The error bound for an alternating series states that the absolute error is less than the next term in the series. By ensuring the next term is less than the desired error margin, we can determine the number of terms needed.⚹◊What is the formula for calculating the error bound in an alternating series according to Theorem 9.9?⍭ Generate me a similar question◊

Question

$\boldsymbol{X}$ Incorrect.

The series converges by the alternating series test. How many terms give a partial sum, $S_{n}$, within 0.02 of the sum, $S$, of the series?
$$
\begin{array}{l}
\sum_{n=1}^{\infty} \frac{(-1)^{n-1}}{n} \
n \geq
\end{array}
$$
i 50
eTextbook and Media

Hint
Assistance Used

Since the series converges and is an alternating series of the form $\sum_{n=1}^{\infty}(-1)^{n-1} a_{n}$, by Theorem 9.9: Error Bounds for Alternating Series you know $\left|S-S_{n}\right|<a_{n+1}$.

Accepted Answer

∻Solution by Steps∻ ‖ step 1 ⋮ To determine the number of terms needed to get a partial sum within 0.02 of the total sum of the series, we use the error bound for alternating series. According to Theorem 9.9, for an alternating series of the form $ \sum_{n=1}^{\infty}(-1)^{n-1} a_{n} $, the error $ \left|S - S_{n}\right| $ is less than $ a_{n+1} $. ‖ ‖ step 2 ⋮ The given series is $ \sum_{n=1}^{\infty} \frac{(-1)^{n-1}}{n} $. Here, $ a_{n} = \frac{1}{n} $. ‖ ‖ step 3 ⋮ To find the number of terms $ n $ such that the error is within 0.02, we need $ a_{n+1} < 0.02 $. ‖ ‖ step 4 ⋮ Since $ a_{n+1} = \frac{1}{n+1} $, we set up the inequality $ \frac{1}{n+1} < 0.02 $. ‖ ‖ step 5 ⋮ Solving for $ n $, we get $ n+1 > 50 $, which simplifies to $ n > 49 $. ‖ ‖ step 6 ⋮ Therefore, we need at least 50 terms to ensure the partial sum is within 0.02 of the total sum. ‖ ∻Answer∻ ⚹ 50 ⚹ ∻Key Concept∻ ⚹Error Bound for Alternating Series⚹ ∻Explanation∻ ⚹The error bound for an alternating series states that the absolute error is less than the next term in the series. By ensuring the next term is less than the desired error margin, we can determine the number of terms needed.⚹◊What is the formula for calculating the error bound in an alternating series according to Theorem 9.9?⍭ Generate me a similar question◊

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