Calculus 2 Exam: Series tests, first order DiffEq

Integrating Factor Form

y’+P(x)y=Q(x), μ(x) = e^(∫P(x) dx)

Integrating Factor Solution

y = (1/μ(x)) [∫ μ(x) Q(x) dx + C]

Geometric Series Form

a_n=arⁿ

Geometric Series Convergence Conditions

converges to (first term/1-r) if |r|<1, diverges if |r|>1

Alternating Series Form

a_n=(-1)ⁿ⁺¹C_n

Alternating Series Test for Convergence

C_n>1 (truly alternating), C_n>C_n+1 (decreasing value being added/subtracted), limC_n=0

Alternating Series Error Estimation

If the series converges via AST, error<C_n+1

Integral Test

if a_n is positive, continuous, decreasing or negative, continuous, increasing, then ∫ a_n and the series of a_n share the same fate

Ratio Test Form

lim(a_n+1/a_n) = p

Ratio Test Conclusions

if 0<p<1, a_n converges absolutely; if p=1, test is inconclusive; if p>0, diverges.

Root Test

Same conclusions as Ratio Test

Direct Comparison Test

o<a_n<b_n where a diverges or b converges, then they share the same fate

Limit Comparison Test

If lim(a_n/b_n)=L (finite), then they share the same fate

Radius of Convergence

Use the ratio/root test to find the limit, find the bounds of convergence (radius = (1/2)(range of bounds), solve if the bounds are included or not