Compute the limit of the partial sums to determine whether the series converges or diverges.
1)6 + 4 - 4 + 4 - 4 + 4 - 4 +...
neither
To determine whether the series converges or diverges, we need to first find the limit of the partial sums.
The given series is an alternating series with the terms switching between positive and negative. We can find the partial sums by considering the pattern:
S1 = 6
S2 = 6 + 4 = 10
S3 = 10 - 4 = 6
S4 = 6 + 4 = 10
S5 = 10 - 4 = 6
From the pattern, we can observe that the partial sums alternate between 6 and 10. Therefore, there is no specific value that the partial sums approach, indicating that the series does not converge.
To verify this, we can express the pattern in a general formula for the nth partial sum:
Sn = 6 + (-1)^(n+1) * 4 = 6 + (-1)^n * 4
Taking the limit as n approaches infinity, we have:
lim (n→∞) (Sn) = lim (n→∞) (6 + (-1)^n * 4)
Now, let's analyze the limit. The term (-1)^n will alternate between -1 and 1 as n goes to infinity. So, the limit becomes:
lim (n→∞) (6 + (-1)^n * 4) = 6 + lim (n→∞) ((-1)^n * 4)
Since the term (-1)^n oscillates between -1 and 1, the limit does not exist. Therefore, the series diverges.
In conclusion, the given series, 6 + 4 - 4 + 4 - 4 + 4 - 4 + ..., diverges.