- 8Mk=0 a0rk
- a/(1-r) if |r|
- Alternating Series Test for Convergence
- Suppose 8Mn=1 (-1)^(n+1)an is an alternating series. If 1)0(lt)a[n+1](lt)a[n] for every n, and lim a[n] = 0, then (-1)^(n+1)converges
- Angle between u and v
- u.v = |u||v| cos(())
- Area through vectors
- A = 1\\2|u||v|sin(()) = 1\\2|uxv|
- Comparison Test - convergence
- M an and M bn are positive-term series, if Mbn converges and an(lt)bn for every n, then an converges
- Comparison Test - divergence
- M an and M bn are positive-term series, if bn diverges and an > bn for every n, then an diverges
- Comparison Test examples
- p/(q+k) < p/q < (p+k)/q, try to simplify to geometric or p-series
- Decreasing function
- if f\'(x) < 0 OR bottom increases while top doesn\'t
- Definition of Absolutely Convergent
- both MAn and M|An| converges
- Definition of Conditionally Convergent
- MAn converges but M|An| diverges (only on Alternating Series and by Limit Comparison or Integral and Alternating Series Test)
- Dot Product of v and u
- u1v1+u2v2+...
- Estimate f(x)
- f(x) = Pn(x) + Rn(x) for some z betseen x and a
- Harmonic Series
- 8Mk=1 1/k = 1+1/2+1/3+1/4+..., which diverges
- Integral Test
- 8Mn=1 an, 1~8 f(x)dx = lim(t->8) 1~t f(x)dx, if 8Mn=1 an is a positive-term series and f is a continous positive-valued, decreasing (eventually) function for eacn n = 1, 2, 3, etc.
- Limit Comparison Test
- Suppose An and Bn are positive-term series;
1)If lim An/Bn = L with 0(lt)L(lt)8, then either both converge or both diverge
2)if lim An\\Bn = 0 and Bn converges, An converges
3) If lim An\\Bn = 8 and Bn diverges, then An diverges - nth degree Taylor Polynomial of f(x) at x = a
- Pn(x) = nMk=0 f(k)(a) * 1/k! * (x-a)k
- nth Term Test for Divergence
- if lim an # 0, then 8Mk=1 an diverges
- p-series
- 8Mn=1 1/np converges if p>1
- Power Series
- 8Mn=0 anxn = A0+A1x+A2x2+...
- ranking of n powers
- ln(n) (lt) n, n^2, n^3 (lt) 2^n, 3^n, 4^n (lt) n!, n^n
- Ratio Test
- p = lim|An+1/An| Then An converges if p1, and test is inconclusive if p=1
- Remainder of nth degree Taylor Polynomial
- Rn(x) = fn+1(z)/(n+1)! * (x-a)n+1
- Root Test
- p = lim |An|1\\n Then An converges if p(lt)1, An diverges if p>1, and test is inconclusive if p=1
- Row Equivalent Operations
- 1)Switch two rows;
2)Multiply any row by constant;
3)Add a row to another - Test for v and u being parallel
- u = cv for real number c
- Test for v and u being perpendicular
- u.v = 0
- Unit Vector
- v is vector. u = v/|v| = v1/|v|, v2/|v|
- Vector Cross Product
- uxv=
|i j k|
|u1 u2 u3|
|v1 v2 v3| - Volume of vectors
- V = |a.(bxc)|
