please show steps for it. the series going from 1-infinity of ((x+2)^n)/(n4^n)
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Use the ratio test.
r = lim(n→∞) | [(x+2)^(n+1) / ((n+1) 4^(n+1))] / [(x+2)^n / (n 4^n)] |
..= |x + 2| * lim(n→∞) n / (4(n+1))
..= |x + 2|/4.
So, this series converges at least for r = |x + 2|/4 < 1 ==> |x + 2| < 4.
Checking the endpoints:
x = 2 ==> Σ(n=1 to ∞) 1/n; divergent p-series.
x = -6 ==> Σ(n=1 to ∞) (-1)^n/n; convergent by the Alternating Series Test, since {1/n} is a decreasing sequence which converges to 0.
So, the interval of convergence is [-6, 2).
I hope this helps!
r = lim(n→∞) | [(x+2)^(n+1) / ((n+1) 4^(n+1))] / [(x+2)^n / (n 4^n)] |
..= |x + 2| * lim(n→∞) n / (4(n+1))
..= |x + 2|/4.
So, this series converges at least for r = |x + 2|/4 < 1 ==> |x + 2| < 4.
Checking the endpoints:
x = 2 ==> Σ(n=1 to ∞) 1/n; divergent p-series.
x = -6 ==> Σ(n=1 to ∞) (-1)^n/n; convergent by the Alternating Series Test, since {1/n} is a decreasing sequence which converges to 0.
So, the interval of convergence is [-6, 2).
I hope this helps!