Find the values of x for which the series converges.
Sigma (lower index n = 0; upper index infinity) [(x+1)/4]^n
To determine the values of x for which the series converges, we can use the concept of geometric series convergence.
A geometric series is given by the formula:
Σ (lower index n = 0; upper index infinity) ar^n
For a geometric series to converge, the absolute value of the common ratio (r) must be less than 1.
In our given series:
Σ (lower index n = 0; upper index infinity) [(x+1)/4]^n
We can observe that the common ratio (r) is (x+1)/4.
Therefore, to find the values of x for which the series converges, we need to find the values of x where the absolute value of (x+1)/4 is less than 1.
|x+1|/4 < 1
To solve this inequality, we will consider two cases:
Case 1: (x+1)/4 < 1
Solving for x in this case:
x + 1 < 4
x < 3
Therefore, for this case, the series converges when x < 3.
Case 2: -(x+1)/4 < 1
Solving for x in this case:
-(x + 1) < 4
x + 1 > -4
x > -5
Therefore, for this case, the series converges when x > -5.
Combining the solutions from both cases, we find that the series converges when -5 < x < 3.