Find the smallest number n of terms needed to obtain an approximation of the series ∞∑ k=1 24ke^(−0.01k)^(2) accurate to 10^−6

Answer 1

For ∑_{(k from 1 to ∞)}[24k×e^(-0.02k)], a programmable calculator will run term-calculation loops until the summation refuses to budge from 59998.00004 (since values of e^(-0.02k) grow extremely tiny). A sample of summations for "k" calculation loops gives:

--k-----------------------------------S--------

100--------------------------35799.78072

200--------------------------54547.15601

300--------------------------58965.82285

700--------------------------59997.25863

1000------------------------59997.99747

1200------------------------59997.99998

1300----------------------- 59998.00003

1327------------------------59998.00004

For accuracy to 1E-6 (that is, 1×10^-6 or one-millionth), one would calculate 59998.00004 × 0.999999

equal to 59997.94004 which places k at approximately 834 as the number of terms needed.