SlytherinPrincess8 SlytherinPrincess8
  • 26-05-2020
  • Mathematics
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Show that the series is convergent by the alternating series test, and find the number of terms necessary to estimate the sum of the series with an error of less than 0.05.

Show that the series is convergent by the alternating series test and find the number of terms necessary to estimate the sum of the series with an error of less class=

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MathPhys
MathPhys MathPhys
  • 26-05-2020

Answer:

6 terms

Step-by-step explanation:

2 − 2/4 + 2/9 − 2/16 + ...

∑ (-1)ⁿ⁺¹ 2 / n²

Applying alternating series test:

lim(n→∞) 2/n² = 0

2/(n+1)² < 2/n², so the series is decreasing.

Therefore, the series converges.

2/(n+1)² < 0.05

(n+1)²/2 > 20

(n+1)² > 40

n+1 > 6.32

n > 5.32

n = 6

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