CALC 2 TEST QUESTIONS AND ANSWERS #11
Limit definition of a series - correct answer write out terms until the pattern is figured
out. Then
Theorem - correct answer if the series is convergent then the limit of sequence is 0
Divergence test - correct answer if limit of the series does not exist or is not 0 then the
series is divergent
Geometric series theorem - correct answer geometric series is convergent if | r | < 1 and
its sum can be found with the equation ar^(n-1)= (a) / (1 - r) if r is greater than or equal
to 1 the series is divergent
Partial fraction decomposition - correct answer seperate the terms with constants over
the denominators. Cross multiply then solve for letters. Using new series from partial
fraction decomp write out terms, then find pattern, create equation and take limit
Integral test - correct answer needs to be continuous, positive, and decreasing. Then if
you take integral of series. The series will converge is integral converges and will
diverge if integral diverges
P series - correct answer the series 1 / n^p is convergent is p > 1 and divergent if p is
less than or equal to 1
Remainder estimate for the integral test - correct answer suppose f(x) is equal to a[n]
where f is a continuous, positive, decreasing function for x gtoe n and a[n] is
convergent. Then the remainder is less than the limit of the integral but more than the
limit of the next term in the integral
Comparison test - correct answer suppose that a[n] and b[n] are series with positive
terms. If b[n] is convergent and a[n] is ltoe b[n] for all n then a[n] is also convergent. If
b[n] is divergent and a[n] is gtoe b[n] for all n then a[n] is also divergent
Limit comparison test - correct answer suppose a[n] and b[n] are series with positive
terms, if limit a[n] / b[n] = c
Where c is a finite number and greater than 0 then either both series converge or both
diverge
Alternating series test - correct answer if the alternating series b[n] satisfies that it is
decreasing and the limit is 0 then the series converges
Alternating series estimation theorem - correct answer if the sum of an alternating seties
satisfies that b[n] is gtoe to b[n+1] and the limit is 0 than r[n] is ltoe to b[n+1]
Limit definition of a series - correct answer write out terms until the pattern is figured
out. Then
Theorem - correct answer if the series is convergent then the limit of sequence is 0
Divergence test - correct answer if limit of the series does not exist or is not 0 then the
series is divergent
Geometric series theorem - correct answer geometric series is convergent if | r | < 1 and
its sum can be found with the equation ar^(n-1)= (a) / (1 - r) if r is greater than or equal
to 1 the series is divergent
Partial fraction decomposition - correct answer seperate the terms with constants over
the denominators. Cross multiply then solve for letters. Using new series from partial
fraction decomp write out terms, then find pattern, create equation and take limit
Integral test - correct answer needs to be continuous, positive, and decreasing. Then if
you take integral of series. The series will converge is integral converges and will
diverge if integral diverges
P series - correct answer the series 1 / n^p is convergent is p > 1 and divergent if p is
less than or equal to 1
Remainder estimate for the integral test - correct answer suppose f(x) is equal to a[n]
where f is a continuous, positive, decreasing function for x gtoe n and a[n] is
convergent. Then the remainder is less than the limit of the integral but more than the
limit of the next term in the integral
Comparison test - correct answer suppose that a[n] and b[n] are series with positive
terms. If b[n] is convergent and a[n] is ltoe b[n] for all n then a[n] is also convergent. If
b[n] is divergent and a[n] is gtoe b[n] for all n then a[n] is also divergent
Limit comparison test - correct answer suppose a[n] and b[n] are series with positive
terms, if limit a[n] / b[n] = c
Where c is a finite number and greater than 0 then either both series converge or both
diverge
Alternating series test - correct answer if the alternating series b[n] satisfies that it is
decreasing and the limit is 0 then the series converges
Alternating series estimation theorem - correct answer if the sum of an alternating seties
satisfies that b[n] is gtoe to b[n+1] and the limit is 0 than r[n] is ltoe to b[n+1]
