Find the first five terms of the sequence of partial sums. (Round your answers to four decimal places.)
∞
∑ (−5)^n + 1/n!
n=1
S1 = 25.000
S2 = -37.5000
S3 ≈ 66.6667
S4 ≈
S5 ≈ 66.6667
Im missing the 4th one, If anyone knows the answer it'd be appreciated.
website marked my previous answers as correct, these can't be it.
To find the fourth term of the sequence of partial sums for the given series, we need to sum up the first four terms of the series.
The series is given by:
∞
∑ (-5)^n + 1/n!
n=1
To find the partial sums, we need to evaluate the series for each value of n up to the desired term.
S1 is the sum of the first term, which is given by:
S1 = (-5)^1 + 1/1! = -5 + 1 = -4
S2 is the sum of the first two terms, which is given by:
S2 = (-5)^1 + 1/1! + (-5)^2 + 1/2! = -4 + 1/2 = -3.5
S3 is the sum of the first three terms, which is given by:
S3 = (-5)^1 + 1/1! + (-5)^2 + 1/2! + (-5)^3 + 1/3! = -4 + 1/2 - 125/6 = -125/6 + 3/6 - 104/6 = -226/6 = -37.6667 ≈ -37.6667
Now, to find the fourth partial sum, we need to sum up the first four terms.
S4 = (-5)^1 + 1/1! + (-5)^2 + 1/2! + (-5)^3 + 1/3! + (-5)^4 + 1/4!
To find the exact value of S4, we need to evaluate this expression. However, since this is an infinite series, it is not possible to find an exact value for S4 without writing out an infinite number of terms.
Therefore, it seems like there might be an error or a pattern in the given terms. It is possible that there was a mistake in the calculation or in providing the information. We would need more information or clarification to determine the exact value of S4.
how did you get your answers?
a_n = (−5)^n + 1/n!
a1 = -5 + 1 = -4
a2 = 25 + 1/2
a3 = -125 + 1/6
a4 = 625 + 1/24
a5 = -3125 + 1/120
So that gives
S1 = -4
S2 = S1 + 25.5555 = 21.5
S3 = S2 -125 + 1/6 = -103.2778
S4 = S3 + 625 + 1/24 = 521.9722
S5 = S4 - 3125 + 1/120 = -2603.0195