Infinite Sequences and Series (BC Only)
What is an effect on the convergence of a series if each term in an alternating series is replaced by , assuming that all original terms were decreasing and positive?
The series converges faster due to the natural logarithm's properties.
Convergence may be affected since doesn't necessarily decrease monotonically for all n.
Convergence isn't affected as long as all terms remain positive.
The series diverges because logarithmic functions have slower decay than polynomials or exponentials.
What general guideline can be given regarding the first error bound when approximating the summation of an infinite series whose terms are increasing positive functions?
The upper bound for the error approximation requires the absolute value of the term immediately succeeding the first omitted term.
The lower bound for the error is provided by using the absolute value of the first omitted term.
The upper bound for the error approximation requires the absolute value of the term succeeding the fifth omitted term.
The lower bound for the error is provided by using the absolute value of the term immediately preceding the first omitted term.
What is the error bound when using the third term of an alternating series to approximate its sum?
Less than or equal to
Greater than or equal to
Greater than or equal to
Less than or equal to
What is true about an alternating series with decreasing positive terms starting with a negative leading coefficient whose nth partial sum exceeds its limit by more than twice the nd-term?
It violates both Leibniz’s test for convergence and established bounds on accuracy based on following terms’ magnitudes indicating divergence or misapplication of tests/bounds.
Incorrect application of Leibniz’s test yet observes bounds rules thus implying potentially slower rate of convergence instead.
There is no violation or implication as long as subsequent terms continue decreasing which guarantees eventual conformance to established thresholds regardless of initial partial sums' exceedance.
Correct application Of Leibniz’s test indicating accelerated rate of convergence beyond expected parameters set By usual bounding principles.
If the series converges, what is the maximum value of when using the Alternating Series Error Bound to approximate the sum with ?
Which of the following best describes the relationship between the error bound and the accuracy of an approximation?
Higher error bound indicates higher accuracy.
Higher error bound indicates lower accuracy.
Lower error bound indicates lower accuracy.
Lower error bound indicates higher accuracy.
If the alternating series has its error bound calculated using the first four terms, what is the maximum possible error?
Less than or equal to
Less than or equal to
Less than or equal to
Less than or equal to

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If the radial coordinate changes sign while the angular coordinate remains fixed, what happens to a point's location?
Not sure where it moves since radial negative values aren't allowed.
It moves across the polar axis into the opposite quadrant.
The point gets closer to the fixed pole but stays in the same quadrant.
There is no change in position since the angle doesn't change.
If an alternating series satisfies for all and , what can we say about using its nth term as an error estimate when approximating its sum?
The error cannot be estimated without knowing exact values of subsequent terms.
The error will be greater than or equal to twice the absolute value of the first omitted term.
The error will be less than or equal to the absolute value of the first omitted term.
The error will oscillate between zero and twice the absolute value of any term.
If an alternating series is convergent, and , what is an upper bound for the error in using to approximate the sum of the series?