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APMA 1110 Test 3 Exam with
complete solutions latest version
When does a sequence converge vs diverge? - CORRECT ANSWER-Converge = when
denominator grows faster than the numerator. When the limit of An = a #, and that
number is finite. Then the sequence converges to that #.
diverge = when numerator grows faster than the denominator or for an alternating series
does not go to one particular value. When the limit of An = infinity, then the sequence
diverges to infinity
How to find the partial sum (An), when given Sn and when top of sigma is infinity vs a
number - CORRECT ANSWER-If top is a number, plug the number into An = Sn and
solve.
If top is infinity, plug number into An = Sn-Sn-1 and solve
How to determine the limit of An - CORRECT ANSWER-Look at the n terms, and divide
if needed.
How to write a rule to find An given Sn? - CORRECT ANSWER-An = Sn - Sn-1
How to determine if something is a geometric series? - CORRECT ANSWER-If it has a
common ratio, r and things are raised to the power of n.
When is a geometric series convergent? divergent? - CORRECT ANSWER-Convergent
|r| < 1
Divergent |r|>1
How to determine Sn - CORRECT ANSWER-a/1-r
r = ratio
a = an with 1 plugged in
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How to find the general formula for a geometric series given the sum - CORRECT
ANSWER-take the first term. That is the "a" value. The find the common ratio (a x r). the
general formula should be axr^n, and when 0(or first term) is plugged in, the a term
should be the value that is calculated
How to find the sum of a geometric series - CORRECT ANSWER-a/1-r
How to determine r for a geometric series given a list of terms - CORRECT ANSWER-
Divide the second term by the first term, third term by the second, etc.
Divergence test - CORRECT ANSWER-if lim(An)≠0, then ∑An is divergent
When to use the Divergence Test - CORRECT ANSWER-When lim An doesn't equal 0
Integral test - CORRECT ANSWER-Where f(x) continuous, positive, decreasing over [1,
∞)and an=f(n)
if 1∫∞f(x) convergent, Σan convergent. If 1∫∞f(x) divergent, Σan divergent.
When can you use the integral test? - CORRECT ANSWER-When the function is
positive, continuous, and decreasing. Write this on the paper!
When completing the integral don't forget to substitute infinity with R
If the limit of An = 0, what does it say about the series? - CORRECT ANSWER-The nth
term test is inconclusive, the series might converge or might diverge
When is a p-series convergent? divergent? - CORRECT ANSWER-Convergent: p>1
Divergent: p<=1
When to use p-series test - CORRECT ANSWER-∑c/n^p. p>1 convergent
Comparison Test - CORRECT ANSWER-Find Bn>An if it converges then An
converges. Find Bn<An if it diverges An diverges.
How to conduct the Comparison test - CORRECT ANSWER-get An (what sigma is
attached to) and Bn (simplified version).
Compare an(non simplified version) to bn(simplified version), 0<= An<=Bn for all n=#
(attached to sigma).
If an< bn, bn diverges and so does an(by comparison test).
if an>bn, an converges by ('test' (like a p-series or geometric series), sigma an is
convergent by comparison test
BRAINSCAPE1
APMA 1110 Test 3 Exam with
complete solutions latest version
When does a sequence converge vs diverge? - CORRECT ANSWER-Converge = when
denominator grows faster than the numerator. When the limit of An = a #, and that
number is finite. Then the sequence converges to that #.
diverge = when numerator grows faster than the denominator or for an alternating series
does not go to one particular value. When the limit of An = infinity, then the sequence
diverges to infinity
How to find the partial sum (An), when given Sn and when top of sigma is infinity vs a
number - CORRECT ANSWER-If top is a number, plug the number into An = Sn and
solve.
If top is infinity, plug number into An = Sn-Sn-1 and solve
How to determine the limit of An - CORRECT ANSWER-Look at the n terms, and divide
if needed.
How to write a rule to find An given Sn? - CORRECT ANSWER-An = Sn - Sn-1
How to determine if something is a geometric series? - CORRECT ANSWER-If it has a
common ratio, r and things are raised to the power of n.
When is a geometric series convergent? divergent? - CORRECT ANSWER-Convergent
|r| < 1
Divergent |r|>1
How to determine Sn - CORRECT ANSWER-a/1-r
r = ratio
a = an with 1 plugged in
BRAINSCAPE1
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How to find the general formula for a geometric series given the sum - CORRECT
ANSWER-take the first term. That is the "a" value. The find the common ratio (a x r). the
general formula should be axr^n, and when 0(or first term) is plugged in, the a term
should be the value that is calculated
How to find the sum of a geometric series - CORRECT ANSWER-a/1-r
How to determine r for a geometric series given a list of terms - CORRECT ANSWER-
Divide the second term by the first term, third term by the second, etc.
Divergence test - CORRECT ANSWER-if lim(An)≠0, then ∑An is divergent
When to use the Divergence Test - CORRECT ANSWER-When lim An doesn't equal 0
Integral test - CORRECT ANSWER-Where f(x) continuous, positive, decreasing over [1,
∞)and an=f(n)
if 1∫∞f(x) convergent, Σan convergent. If 1∫∞f(x) divergent, Σan divergent.
When can you use the integral test? - CORRECT ANSWER-When the function is
positive, continuous, and decreasing. Write this on the paper!
When completing the integral don't forget to substitute infinity with R
If the limit of An = 0, what does it say about the series? - CORRECT ANSWER-The nth
term test is inconclusive, the series might converge or might diverge
When is a p-series convergent? divergent? - CORRECT ANSWER-Convergent: p>1
Divergent: p<=1
When to use p-series test - CORRECT ANSWER-∑c/n^p. p>1 convergent
Comparison Test - CORRECT ANSWER-Find Bn>An if it converges then An
converges. Find Bn<An if it diverges An diverges.
How to conduct the Comparison test - CORRECT ANSWER-get An (what sigma is
attached to) and Bn (simplified version).
Compare an(non simplified version) to bn(simplified version), 0<= An<=Bn for all n=#
(attached to sigma).
If an< bn, bn diverges and so does an(by comparison test).
if an>bn, an converges by ('test' (like a p-series or geometric series), sigma an is
convergent by comparison test
BRAINSCAPE1
