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.
you already asked this as Sushi.
See your post below, and have a little patience.
To find the fourth term of the sequence of partial sums for the given series, we need to evaluate the sum from n = 1 to 4. The general formula for the nth partial sum of the given series is:
Sn = ∑ ((-5)^(n-1) + 1/n!), from n = 1 to n
Let's calculate the values step by step:
S1 = (-5)^(1-1) + 1/1! = 1 + 1 = 2.0000
S2 = (-5)^(2-1) + 1/2! + (-5)^(1-1) + 1/1! = -5 + 0.5000 + 1 + 1 = -2.5000
S3 = (-5)^(3-1) + 1/3! + (-5)^(2-1) + 1/2! + (-5)^(1-1) + 1/1! = 25 + 0.1667 - 5 + 0.5000 + 1 + 1 = 22.6667
S4 = (-5)^(4-1) + 1/4! + (-5)^(3-1) + 1/3! + (-5)^(2-1) + 1/2! + (-5)^(1-1) + 1/1!
Now, let's plug in the values and calculate S4:
S4 = (-5)^(4-1) + 1/4! + (-5)^(3-1) + 1/3! + (-5)^(2-1) + 1/2! + (-5)^(1-1) + 1/1!
= (-5)^3 + 1/4! + (-5)^2 + 1/3! + (-5)^1 + 1/2! + (-5)^0 + 1/1!
= -125 + 1/24 + 25 + 1/6 + (-5) + 1/2 + 1 + 1
= -99.7917
Hence, the fourth term of the sequence of partial sums for the given series is approximately -99.7917 when rounded to four decimal places.