Let's have a look at the alternating geometric series $$\sum_{n=0}^{\infty}\left (-\frac{1}{2}\right)^{n}.$$ What's the alternating series estimate for the difference between the sum of these series and $$1-\frac{1}{2}+\frac{1}{4}-\frac{1}{8}+\frac{1}{16}?$$ (A positive fraction, input as a/b in lowest terms.)

Question

BlackTom AI · Accepted Answer

We start by identifying the two series involved. The infinite geometric series is S = ∑_{n=0}^∞ (-1/2)^n, and the finite partial sum given is S4 = 1 - 1/2 + 1/4 - 1/8 + 1/16, which corresponds to the sum up to n = 4.

First, compute the actual sums: the infinite sum of a geometric series with ratio r = -1/2 is S = 1 / (1 - r) = 1 / (1 + 1/2) = 1 / (3/2) = 2/3. The partial sum S4 can be evaluated step by step: 1 - 1/2 = 1/2, +1/4 = 3/4, -1/8 = 5/8, +1/16 = 11/16. So S4 = 11/16.

Now look at the difference R4 = S − S4. Since the series is alternating with decreasing term magnitudes, the Alternating Series Test gives a standard remainder bound: the magnitude of the remainder is at most the magnitude of the next term, and the remainder has the same sign as the next term. The next term (n = 5) is (-1/2)^5 = -1/32, so the remainder R4 lies between 0 and -1/32. In magnitude, |R4| ≤ 1/32.

To get a sense of the actual difference, compute S − S4 explicitly: 2/3 − 11/16. Using a common denominator of 48, this is 32/48 − 33/48 = -1/48. This value lies within the predicted interval [−1/32, 0], consistent with the alternating-series remainder behavior. The standard short estimate for the magnitude of the remainder is given by the next term’s magnitude, which is 1/32.

Therefore, the alternating-series estimate for the difference has magnitude 1/32. The exact remainder is -1/48, which confirms the bound and shows the estimate is an upper bound in magnitude rather than an exact value.

Key takeaways:
- S = 2/3 for the infinite alternating geometric series with ratio -1/2.
- S4 = 11/16 for the given five-term partial sum.
- The remainder R4 satisfies 0 > R4 > -1/32, so |R4| ≤ 1/32, and the standard estimate uses 1/32 as the bound.
- The exact difference is -1/48, illustrating that the actual remainder is smaller in magnitude than the bound 1/32.

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