Prove series 2, (-1/2), (2/9), (-1/8) is convergent by the alternating series test and find the...

Prove series 2, (-1/2), (2/9), (-1/8) is convergent by the alternating series test and find the number of terms required to estimate the sum of the series with an error of less than 0.05

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milcah answered 2 years ago

Find the values of p for which the series is convergent (a) Σ(n=2 to ∞) 1/n(ln...

Find the values of p for which the series is convergent (a) Σ(n=2 to ∞) 1/n(ln n)^p (b) Σ(n=2 to ∞) n(1+n^2)^p HINT: Use the integral test to investigate both

\sum _{n=1}^{\infty }\:\frac{4\left(-1\right)^n+2^n}{3^n} Determine whether the series is convergent or divergent. If it is convergent, find...

\sum _{n=1}^{\infty }\:\frac{4\left(-1\right)^n+2^n}{3^n} Determine whether the series is convergent or divergent. If it is convergent, find its sum.

In this exercise, we examine one of the conditions of the Alternating Series Test. Consider the...

In this exercise, we examine one of the conditions of the Alternating Series Test. Consider the alternating series 1−1+1/2−1/4+1/3−1/9+1/4−1/16+⋯, where the terms are selected alternately from the sequences {1/n} and {−1/n^2}. Explain why the nth term of the given series converges to 0 as n goes to infinity. Rewrite the given series by grouping terms in the following manner: (1−1)+(1/2−1/4)+(1/3−1/9)+(1/4−1/16)+⋯. Use this regrouping to determine if the series converges or diverges. Explain why the condition that the sequence {an}{an} decreases...

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Alternating Series Test. Let (an) be a sequence satisfying (i) a1 ≥ a2 ≥ a3 ≥...

Alternating Series Test. Let (an) be a sequence satisfying (i) a1 ≥ a2 ≥ a3 ≥ · · · ≥ an ≥ an+1 ≥ · · · and (ii) (an) → 0. Show that then the alternating series X∞ n=1 (−1)n+1an converges using the following two different approaches. (a) Show that the sequence (sn) of partial sums, sn = a1 − a2 + a3 − · · · ± an is a Cauchy sequence Alternating Series Test. Let (an) be...

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