MATH 2414 FINAL EXAM 3+ MATERIAL

MATH 2414 FINAL EXAM 3+ MATERIAL



P-Test---ans-✔✔Σ-1/x^p
-Converges-is-p->-1
(Can-subtract-degree-of-P(x)-&-G(x)-to-simplify

Geometric-Seriea---ans-✔✔Σ-an*r^n
-|-r-|-<-1,-Series-converges
-Sum-=-a1/(1-r)

Alternating-Series-Test---ans-✔✔Σ-(-1)^n*bn
-if-bn->-0-&-decreasing
-If-Lim-[n-->-oo]-of-bn-=-0,-Series-converges

Limit-Comparison-Test---ans-✔✔If-Lim-[n-->-oo]-of-an/bn-=-C-(a-finite-number),-then-the-
convergence-of-an-&-bn-are-equivalent

Ratio-Test---ans-✔✔-If-Lim-[n-->-oo]-of-|-an+1/an-|-=-L-<1,-then-Σan-is-absolutely-
convergent
-If-Lim-[n-->-oo]-of-|-an+1/an-|-=-L->1-or-=-oo,-then-Σan-is-divergent
-If-Lim-[n-->-oo]-of-|-an+1/an-|-=-L-=1-or-DNE,-the-test-is-inconclusive
*Best-used-to-cancel-factorials-or-with-power-series*

Root-Test---ans-✔✔-If-Lim-[n-->-oo]-of-nrt(|-an-|)-=-L-<1,-then-Σan-is-absolutely-convergent
-If-Lim-[n-->-oo]-of-nrt(|-an-|)-=-L->1-or-=-oo,-then-Σan-is-divergent
-If-Lim-[n-->-oo]-of-nrt(|-an-|)-=-L-=1-or-DNE,-the-test-is-inconclusive

Integral-Test---ans-✔✔If-f(n)->-0-and-decreasing-from-1-to-oo,-let-an-=-f(n),-then-the-series-
converges-or-diverges-as-∫-[1-to-oo]-f(n)-does

Integral-Test-Error-Estimate---ans-✔✔•∫-[n+1-to-oo]-of-f(n)-<=-Rn-<=-∫-[n-to-oo]-of-f(n)
•∫-[n+1-to-oo]-of-f(n)-+-Sn-<=-S-<=-∫-[n-to-oo]-of-f(n)-+-Sn

Alternating-Series-Error-Estimate---ans-✔✔•|-Rn-|-<-b_n+1
Set-given-Rn-=-bn-and-solve-for-n-at-which-that-error-occurs.-Final-answer-is-that-n---1

Power-Series---ans-✔✔Σ-[n=0]-of-Cn-(x-a)^n
•a-is-series-center
•use-ratio-test-to-determine-radius-of-convergence-and-interval-of-convergence