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1
Q
lim x->0 sinx/x =
A
= 1
2
Q
lim x->0 1-cosx/x² =
A
= 1/2
3
Q
lim x->∞ (1+1/x)² =
A
= e
4
Q
lim x-> -∞ (1+1/x)²=
A
= e
5
Q
lim x->0 log(1+x)/x =
A
= 1
6
Q
lim x-> 0 eˣ -1 / x =
A
= 1
7
Q
lim x->∞ eˣ/xᵇ
A
= ∞
8
Q
lim x->∞ logx/xᵇ =
A
= 0
9
Q
lim x->0+ xᵇlogx =
A
= 0
10
Q
d/dx logx
A
1/x
11
Q
d/dx sinx
A
cosx
12
Q
d/dx cosx
A
-sinx
13
Q
d/dx tanx
A
sec²x
14
Q
d/dx cotx
A
-cosec²x
15
Q
d/dx arcsinx
A
1/√(1-x²)
16
Q
d/dx arccosx
A
-1/√(1-x²)
17
Q
d/dx arctanx
A
1/√(1+x²)
18
Q
d/dx sinhx
A
coshx
19
Q
d/dx coshx
A
sinhx
20
Q
d/ dx arccoshx
A
1/√(x²-1)
21
Q
Sequence Convergence Tests
A
- Null sequence test
- The comparison test
- The ratio test
- The root test
- The Integral test
- Alternating Series Test
- Absolute Convergence Test
22
Q
Null Sequence Test
A
if Σaₙ converges, then an->0
test for divergence only: if an->0 then Σaₙ diverges
23
Q
The Comparison Test
A
Suppose an>=0 bn>=an for n then:
- Σbₙ converges <=> Σaₙ converges
- Σaₙ diverges <=> Σbₙ diverges
24
Q
The Ratio Test
A
Suppose Σaₙ is a series of non-negative terms and: aₙ₊₁/aₙ -> r then: 1. if r<1 => Σaₙ converges 2. if r>1 Σaₙ diverges 3. if r=0 inconclusive
25
The root test
Suppose Σaₙ is a series of non-negative terms and aₙ¹/ⁿ->. Then:
1. if r<1 => Σaₙ converges
2. if r>1 => Σaₙ diverges
3. if r=0 inconclusive
26
Integral Test
Suppose f:[1,∞)-> is continuous, decreasing and positive in its domain. Then the series Σfₙ converges if and only if the sequence (∫f(x)dx) converges
27
Alternating series test
Consider the series Σ(-1)ⁿ⁺¹aₙ where the sequence (aₙ) is decreasing and converging to zero. Then the series converges.
28
Absolute Convergence Test
If a series converges absolutely absolutely, then it converges
