The error is less than the magnitude of the first term omitted, which is l/ll
2 =0.0083.
37.60
Estimate the error when
is approximated by its first 10 terms.
We must find the least n such that 1/4" < 0.005= 255, that is, 200 < 4", or n>4. So, if we use
l-s + Tg-s = t4, the error will be less than 0.0005. Since H = 0.796 • • •, our approximation is 0.80.
To check, note that the given series is a geometric series with ratio - \, and, therefore, the sum of the series is
Thus, our approximation is actually the exact value of the sum.
37.59
Approximate the sum
the alternating series test. Check your answer by finding the actual sum.
to two decimal places using the method based on
We want an error < 0.0005. Hence, we must find the least n so that
that is,
n!>2000. Since 6! =720 and 7! = 5040, the desired value of n is 7. So we must use -l+^-£ +
s-Tio + 73o=-7S~ -0.628. [The actual value is e~
l - 1.]
37.58
Find the sum of the infinite series
correct to three decimal places.
The error is less than the magnitude of the first term omitted. Thus, the approximation is 1 — k + 3 =
|, and the error is less than j. Hence, the actual value V satisfies n<^
37.57
Find the error if the sum of the first three terms is used as an approximation to the sum of the alternating series
Although the terms are alternatively positive and negative, the alternating series test does not apply, since
Since the nth term does not approach 0, the series is not convergent.
37.56
Determine whether
converges.
By the ratio test, the series is absolutely convergent; so it is certainly convergent.
37.55
Determine whether
is convergent.
Use the comparison test with the convergent p-series
Clearly,
Hence, the
given series converges. (The integral test is also applicable.)
37.54
Determine whether
is convergent.
Hence, the series converges. (The integral test is also applicable.)
Apply the ratio test.
37.53
Determine whether
is convergent
Case I. If
Cose //. If
Case III. If
Assume
for
the series is absolutely convergent,
the series is divergent,
nothing can be said about convergence or divergence.
318
CHAPTER 37
37.52
State the ratio test for a series E a n .
2 =0.0083.
37.60
Estimate the error when
is approximated by its first 10 terms.
We must find the least n such that 1/4" < 0.005= 255, that is, 200 < 4", or n>4. So, if we use
l-s + Tg-s = t4, the error will be less than 0.0005. Since H = 0.796 • • •, our approximation is 0.80.
To check, note that the given series is a geometric series with ratio - \, and, therefore, the sum of the series is
Thus, our approximation is actually the exact value of the sum.
37.59
Approximate the sum
the alternating series test. Check your answer by finding the actual sum.
to two decimal places using the method based on
We want an error < 0.0005. Hence, we must find the least n so that
that is,
n!>2000. Since 6! =720 and 7! = 5040, the desired value of n is 7. So we must use -l+^-£ +
s-Tio + 73o=-7S~ -0.628. [The actual value is e~
l - 1.]
37.58
Find the sum of the infinite series
correct to three decimal places.
The error is less than the magnitude of the first term omitted. Thus, the approximation is 1 — k + 3 =
|, and the error is less than j. Hence, the actual value V satisfies n<^
37.57
Find the error if the sum of the first three terms is used as an approximation to the sum of the alternating series
Although the terms are alternatively positive and negative, the alternating series test does not apply, since
Since the nth term does not approach 0, the series is not convergent.
37.56
Determine whether
converges.
By the ratio test, the series is absolutely convergent; so it is certainly convergent.
37.55
Determine whether
is convergent.
Use the comparison test with the convergent p-series
Clearly,
Hence, the
given series converges. (The integral test is also applicable.)
37.54
Determine whether
is convergent.
Hence, the series converges. (The integral test is also applicable.)
Apply the ratio test.
37.53
Determine whether
is convergent
Case I. If
Cose //. If
Case III. If
Assume
for
the series is absolutely convergent,
the series is divergent,
nothing can be said about convergence or divergence.
318
CHAPTER 37
37.52
State the ratio test for a series E a n .