Question

If the series is convergent, use the Alternating Series Estimation Theorem to determine how many terms we need to add in order to find the sum with an error less than 0.000005. (If the quantity diverges, enter DIVERGES.) 6 Incorrect: Your answer is incorrect. terms Need Help? Read It Watch It

Answer 1

To determine the number of terms needed to find the sum with an error less than 0.000005, use the Alternating Series Estimation Theorem and find the smallest value of n such that |an| < 0.000005.

Step-by-step explanation:

To determine the number of terms we need to add in order to find the sum with an error less than 0.000005, we can use the Alternating Series Estimation Theorem. This theorem states that if an alternating series converges, the absolute value of the error between the sum of the series and the actual value is less than or equal to the absolute value of the next term. In other words, we need to find the smallest value of n such that |an+1| < 0.000005.

Let's say the series is given by S = a1 - a2 + a3 - a4 + a5 - a6 + ...

According to the Alternating Series Estimation Theorem, |an+1| < 0.000005 means that |an| < 0.000005.

Therefore, we need to find the smallest value of n such that |an| < 0.000005. Once we find this value, we add up the first n terms to get an approximation of the sum with an error less than 0.000005.

Answer 2

Number of terms required is 5 terms

Step-by-step explanation:

Given the series:
`\sum\imits^(\infty)_(n=0)((-1)^n)/(5^nn!)`

Given the series is an alternating series,

`b_n=(1)/(5^nn!)`

Evaluating the limit:
`\lim_(n \to \infty) b_n= \lim_(n \to \infty) ((1)/(5^nn!))=(1)/(\infty)=0`

Since
`\lim_(n \to \infty) b_n =0 \,\, and \,\, b_(n+1)\leq b_n` for all n = the series is convergent

The error of an alternating series
`\sum b_n` is bounded as

`|R_n|\leq b_(n+1)`

Given
`b_n = (1)/(5^n n!)=b_(n+1)=(1)/(5^(n+1)(n+1)!)<0.000005\\\\=5^(n+1)(n+1)!>200000`

By trial and error: the above equation is satisfied for
`n=4`

Since the given series starts at n = 0, the number of terms required is 5 terms