CALC 2 FINAL EXAM WITH SOLUTIONS #19
Divergence test - correct answer if ∞∑i=1 ai may converge ONLY if lim i→∞ ai=0
Integral test - correct answer if ∫1→∞ f(x)dx = ∞ then ∞∑i=1 ai is divergent
If ∫1→∞ f(x)dx < ∞ then∞∑i=1 ai is convergent
Geometric series test - correct answer if {ai} is a geometric sequence and if |r|<1 then
∞∑i=1 ai converges
P-series test - correct answer ∞∑i=1 1/i^p
P≤1 diverges
P>1 converges
Ratio test - correct answer (FAILS WHEN R=1)
Supposed ai ≥ 0, lim (ai+1)/(ai) = r then ∞∑i=1 ai converges when r<1 and diverges
when r>1
Comparison test - correct answer supposed we have two non-negative series ∞∑i=1 ai
and ∞∑i=1 bi
- if ai≥0 and bi≥0 and ai>bi for all i then if bi diverges ai also diverges
- if ai>bi for all i and ai converges then bi also converges
Limit comparison test - correct answer supposed ai≥0, bi≥0 and 0<limi→∞ ai/bi<∞ then
they both either diverge or converge. They are the same
Alternating Series test - correct answer ∞∑n=0 (-1)^n*an
An≥0
If an→0 as n→∞ and alternating sign then it converges
Conditional convergence v. Absolutely convergence - correct answer ∞∑i=1 ai is said to
converge absolutely if ∞∑i=1 |ai| converges.
Absolute convergence implies conditional convergence but the appositive is not true
Error analysis for the alternating series test - correct answer n∑i=1 (-1)^i*Ai
E<An+1
Error analysis for convergent alternating series example - correct answer 1/(i+1)² < 0.1
(i+1)² > 10
I+1 > √10
I > √10 -1
I=3
Power series at x=0 using a_i=f^(i)(0)/i! - correct answer f(x) =
a₀+a₁x+a₂x²+a₃x³+a₄x⁴+....
F'(x) = a₁+2a₂x+3a₃x²+4a₄x³+.....
, F''(x) = 2a₂+6a₃x+12a₄x²+....
Power series at x=a ( not necessarily 0) using a_i = a_i=f^(i)(a)/i! - correct answer f(x) =
a₀+a₁(x-c)+a₂(x-c)²+a₃(x-c)³+a₄(x-c)⁴+....
F'(x) = a₁+2a₂(x-c)+3a₃(x-c)²+4a₄(x-c)³+.....
F''(x) = 2a₂+6a₃(x-c)+12a₄(x-c)²+....
Radius of convergence of a power series - correct answer |x| < 1
X=1 or x=-1 divergent
-1<x<1
Based on ratio test
Arthmetric sum formula - correct answer Sn = n [(a1+an)/2]
Geometric sum formula - correct answer Sn = ai[(rⁿ-1)/(r-1)]
Infinite geometric sum formula - correct answer S∞= ai/(1-r)
ONLY if |r| < 1
(2+i+3)/[3^(i+1)] * (3^i)/(2+i) - correct answer 3i and 3^(i+1) cancel
[2^(i+1)]/[(i+1)!] * (i!)/(2+^i) - correct answer i! And (i+1)! Cancel
To add all terms in a series... - correct answer the series must be convergent
Finite series... - correct answer 1→ a number (not ∞)
Neither convergent or divergent
Series approximation for e^x - correct answer ∞∑i=0 (x^i)/i!
Series approximation for sinx - correct answer ∞∑i=0 (-1)^i * [x^(2i+1)/(2i+1)!]
Series approximation for cosx - correct answer ∞∑i=0 (-1)^i * [(x^2i)/(2i)!]
Series approximation for 1/1-x - correct answer ∞∑i=0 x^i
Series approximation for 1/(1-x)² - correct answer ∞∑i=0 (1+i)*x^i
In a geometric series when i=0 - correct answer add 1 to n because i=0 not 1
But use 0 to find ai
Average rate of change - correct answer [f(x+h)-f(x)]/h
Fundamental theorem of calculus - correct answer ∫(a-b) f(x)dx=F(b)-F(a)
Where F'=f(x)
Divergence test - correct answer if ∞∑i=1 ai may converge ONLY if lim i→∞ ai=0
Integral test - correct answer if ∫1→∞ f(x)dx = ∞ then ∞∑i=1 ai is divergent
If ∫1→∞ f(x)dx < ∞ then∞∑i=1 ai is convergent
Geometric series test - correct answer if {ai} is a geometric sequence and if |r|<1 then
∞∑i=1 ai converges
P-series test - correct answer ∞∑i=1 1/i^p
P≤1 diverges
P>1 converges
Ratio test - correct answer (FAILS WHEN R=1)
Supposed ai ≥ 0, lim (ai+1)/(ai) = r then ∞∑i=1 ai converges when r<1 and diverges
when r>1
Comparison test - correct answer supposed we have two non-negative series ∞∑i=1 ai
and ∞∑i=1 bi
- if ai≥0 and bi≥0 and ai>bi for all i then if bi diverges ai also diverges
- if ai>bi for all i and ai converges then bi also converges
Limit comparison test - correct answer supposed ai≥0, bi≥0 and 0<limi→∞ ai/bi<∞ then
they both either diverge or converge. They are the same
Alternating Series test - correct answer ∞∑n=0 (-1)^n*an
An≥0
If an→0 as n→∞ and alternating sign then it converges
Conditional convergence v. Absolutely convergence - correct answer ∞∑i=1 ai is said to
converge absolutely if ∞∑i=1 |ai| converges.
Absolute convergence implies conditional convergence but the appositive is not true
Error analysis for the alternating series test - correct answer n∑i=1 (-1)^i*Ai
E<An+1
Error analysis for convergent alternating series example - correct answer 1/(i+1)² < 0.1
(i+1)² > 10
I+1 > √10
I > √10 -1
I=3
Power series at x=0 using a_i=f^(i)(0)/i! - correct answer f(x) =
a₀+a₁x+a₂x²+a₃x³+a₄x⁴+....
F'(x) = a₁+2a₂x+3a₃x²+4a₄x³+.....
, F''(x) = 2a₂+6a₃x+12a₄x²+....
Power series at x=a ( not necessarily 0) using a_i = a_i=f^(i)(a)/i! - correct answer f(x) =
a₀+a₁(x-c)+a₂(x-c)²+a₃(x-c)³+a₄(x-c)⁴+....
F'(x) = a₁+2a₂(x-c)+3a₃(x-c)²+4a₄(x-c)³+.....
F''(x) = 2a₂+6a₃(x-c)+12a₄(x-c)²+....
Radius of convergence of a power series - correct answer |x| < 1
X=1 or x=-1 divergent
-1<x<1
Based on ratio test
Arthmetric sum formula - correct answer Sn = n [(a1+an)/2]
Geometric sum formula - correct answer Sn = ai[(rⁿ-1)/(r-1)]
Infinite geometric sum formula - correct answer S∞= ai/(1-r)
ONLY if |r| < 1
(2+i+3)/[3^(i+1)] * (3^i)/(2+i) - correct answer 3i and 3^(i+1) cancel
[2^(i+1)]/[(i+1)!] * (i!)/(2+^i) - correct answer i! And (i+1)! Cancel
To add all terms in a series... - correct answer the series must be convergent
Finite series... - correct answer 1→ a number (not ∞)
Neither convergent or divergent
Series approximation for e^x - correct answer ∞∑i=0 (x^i)/i!
Series approximation for sinx - correct answer ∞∑i=0 (-1)^i * [x^(2i+1)/(2i+1)!]
Series approximation for cosx - correct answer ∞∑i=0 (-1)^i * [(x^2i)/(2i)!]
Series approximation for 1/1-x - correct answer ∞∑i=0 x^i
Series approximation for 1/(1-x)² - correct answer ∞∑i=0 (1+i)*x^i
In a geometric series when i=0 - correct answer add 1 to n because i=0 not 1
But use 0 to find ai
Average rate of change - correct answer [f(x+h)-f(x)]/h
Fundamental theorem of calculus - correct answer ∫(a-b) f(x)dx=F(b)-F(a)
Where F'=f(x)
