CALCULUS II SERIES TESTS (GEOMETRIC SERIES) WITH
ANSWERS #8
Telescoping Series - correct answer A series whose partial sums eventually only have a
fixed number of terms after cancellation. Write out first several terms and see what does
not cancel.
Divergence Test - correct answer ∑An diverges if lim(An)≠0. Does not imply
convergence. Convergence is inconclusive via the divergence test.
Harmonic series - correct answer Σ 1/n from 1 to infinity.
Always divergent.
Combination of series - correct answer Σ from 1 to infinity of an and bn are both
convergent series Σ(an+bn) and ΣC*an are also convergent
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.
P-Series test - correct answer 1/n^p converges if p >1 and diverges if p =< 1
Direct Comparison Test - correct answer If An < Bn, then An converges if Bn converges;
If An < Bn, then Bn diverges if An diverges
Limit Comparison Test - correct answer an>0, bn>0
if lim n approaches ∞ of an/bn = C (where C is finite and positive), then both series will
either converge or diverge
Alternating Series Test - correct answer if series is alternating i.e Σ(-1)^n+1
if Bn is positive
bn's are decreasing; Bn+1 <= Bn
limBn --> 0
series converges. if not, it diverges
The Alternating Series Estimation Theorem - correct answer If sum S= Σ(-1)^n-1 * bn
satisfies the conditions in the AST as in sequence is decreasing bn+1 <= bn and the lim
n--> ∞ bn = 0 , then we can estimate
|Rn| = |S-Sn| <= bn+1 where R is the remainder after n terms. S is the true sum and Sn
is the partial sum
Absolute Convergence - correct answer If ∑|an| converges then ∑(-1)^n * an is
absolutely convergent
Ratio Test - correct answer For ∑an (a general or alternating series) if lim n --> ∞ |
an+1/an | = L
ANSWERS #8
Telescoping Series - correct answer A series whose partial sums eventually only have a
fixed number of terms after cancellation. Write out first several terms and see what does
not cancel.
Divergence Test - correct answer ∑An diverges if lim(An)≠0. Does not imply
convergence. Convergence is inconclusive via the divergence test.
Harmonic series - correct answer Σ 1/n from 1 to infinity.
Always divergent.
Combination of series - correct answer Σ from 1 to infinity of an and bn are both
convergent series Σ(an+bn) and ΣC*an are also convergent
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.
P-Series test - correct answer 1/n^p converges if p >1 and diverges if p =< 1
Direct Comparison Test - correct answer If An < Bn, then An converges if Bn converges;
If An < Bn, then Bn diverges if An diverges
Limit Comparison Test - correct answer an>0, bn>0
if lim n approaches ∞ of an/bn = C (where C is finite and positive), then both series will
either converge or diverge
Alternating Series Test - correct answer if series is alternating i.e Σ(-1)^n+1
if Bn is positive
bn's are decreasing; Bn+1 <= Bn
limBn --> 0
series converges. if not, it diverges
The Alternating Series Estimation Theorem - correct answer If sum S= Σ(-1)^n-1 * bn
satisfies the conditions in the AST as in sequence is decreasing bn+1 <= bn and the lim
n--> ∞ bn = 0 , then we can estimate
|Rn| = |S-Sn| <= bn+1 where R is the remainder after n terms. S is the true sum and Sn
is the partial sum
Absolute Convergence - correct answer If ∑|an| converges then ∑(-1)^n * an is
absolutely convergent
Ratio Test - correct answer For ∑an (a general or alternating series) if lim n --> ∞ |
an+1/an | = L
