Fig. 37-1
37.67 What is the error if we approximate a convergent geometric series
The error is ar" + ar"*
1 + • • •, a geometric series with sum ar"l(\ - r).
INFINITE SERIES
319
37.61 How many terms must be used to approximate the sum of the series in Problem 37.60 correctly to one decimal
place?
We must have l/rc
2 <0.05 = 55, or 20
5 ~ la — iif =0.79. Hence, correct to one decimal place, the sum is 0.8.
37.62 Estimate the error when the series 1— 5 + s — $ + • •• is approximated by its first 50 terms.
a n = (-l)"
+I /(2« - 1). The error will be less than the magnitude of the first term omitted: 1/[2(51) - 1] =
TOT . Notice how poor the guaranteed approximation is for such a large number of terms.
37.63 Estimate the number of terms of the series
which will be correct to four decimal places.
required for an approximation of the sum
We must have l/n" < 0.00005 = 55300, n"> 20,000, n > 12. Hence, we should use the first 11 terms.
37.64 Show that, if £ a n converges by the integral test for a function f(x), the error R n , if we use the first n terms,
satisfies
If we approximate by the lower rectangles in Fig. 37-1, then
If we use the upper rectangles,
37.65 Estimate the error when
is approximated by the first 10 terms.
By Problem 37.64,
Hence, the error lies between 0.023 and
In addition,
37.66
How many terms are necessary to approximate
correctly to three decimal places?
By Problem 37.64, the error
we need 100 < n2, n>10. So at least 11 terms are required.
To get
by the first n terms a+ar+...+
arn-1?
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from Wow! eBook
37.67 What is the error if we approximate a convergent geometric series
The error is ar" + ar"*
1 + • • •, a geometric series with sum ar"l(\ - r).
INFINITE SERIES
319
37.61 How many terms must be used to approximate the sum of the series in Problem 37.60 correctly to one decimal
place?
We must have l/rc
2 <0.05 = 55, or 20
5 ~ la — iif =0.79. Hence, correct to one decimal place, the sum is 0.8.
37.62 Estimate the error when the series 1— 5 + s — $ + • •• is approximated by its first 50 terms.
a n = (-l)"
+I /(2« - 1). The error will be less than the magnitude of the first term omitted: 1/[2(51) - 1] =
TOT . Notice how poor the guaranteed approximation is for such a large number of terms.
37.63 Estimate the number of terms of the series
which will be correct to four decimal places.
required for an approximation of the sum
We must have l/n" < 0.00005 = 55300, n"> 20,000, n > 12. Hence, we should use the first 11 terms.
37.64 Show that, if £ a n converges by the integral test for a function f(x), the error R n , if we use the first n terms,
satisfies
If we approximate by the lower rectangles in Fig. 37-1, then
If we use the upper rectangles,
37.65 Estimate the error when
is approximated by the first 10 terms.
By Problem 37.64,
Hence, the error lies between 0.023 and
In addition,
37.66
How many terms are necessary to approximate
correctly to three decimal places?
By Problem 37.64, the error
we need 100 < n2, n>10. So at least 11 terms are required.
To get
by the first n terms a+ar+...+
arn-1?
Download
from Wow! eBook