Course: Accelerated Engineering Calculus II Instructor: Michael Medvinsky Handout 2 ∞
Definition: A sum of infinite sequence {an }n=1 , S = ∑ a j is called (infinite) series. ∞
j=1
∞
Definition: In order to find a sum of infinite sequence S = ∑ a j one defines a sequence of j=1
∞
⎧⎪ n ⎫⎪
partial sums as {Sn }n=1 = ⎨∑ a j ⎬ . If the sequence is convergent then S = lim Sn and the series n→∞
⎪⎩ j=1 ⎪⎭n=1
∞
is called convergent. Otherwise the series is divergent. ∞
Theorem: If S = ∑ an is convergent series, then n=1
lim an = lim ( Sn+1 − Sn ) = lim Sn+1 − lim Sn = S − S = 0 n→∞
n→∞
n→∞
x n→∞
Note: The converse theorem doesn’t true. ∞
The test for Divergence: If lim an ≠ 0 or DNE then S = ∑ an is divergent. n→∞
n=1
∞
∞
∞
n=1
n=1
n=1
Theorem: If ∑ an and ∑ bn are convergent series, then so are the series ∑ can (c is constant) ∞
∞
∞
∞
∞
∞
n=1
n=1
n=1
n=1
n=1
n=1
and ∑ ( an ± bn ) . Furthermore: ∑ can = c∑ an and ∑ ( an ± bn ) = ∑ an ± ∑ bn . The Integral test: Let f be continues, positive decreasing function on [ m,∞ ) and let
an = f ( n ) for n ≥ m then the series
∞
∞
n=m
m
∑ an converges if and only if
∫ f ( x ) dx .
Note: Since finite number of terms cannot affect convergence of infinite series, it is enough that f ( x ) decreasing on [ M ,∞ ) where M > m . Course: Accelerated Engineering Calculus II Instructor: Michael Medvinsky The Comparison Test: Let ∑ an , ∑ bn be series with an ,bn > 0 , then -­‐ If
∑b
is convergent and an ≤ bn then
-­‐ If
∑b
is divergent and an ≥ bn then
n
n
∑a
n
∑a
n
is also convergent
is also divergent
The Limit Comparison Test: Let ∑ an , ∑ bn be series with an ,bn > 0 . If lim
n→∞
an
= c where c > 0
bn
is a finite constant, then either both series converge or both diverge. Definition: Let an > 0 , then ∞
∞
∑ b = ∑ ( −1)
n
n=1
n=1
∞
∞
∑ c = ∑ ( −1)
n
n=1
n
an = −a1 + a2 − a3 + a4 − +...
n−1
an = a1 − a2 + a3 − a4 + −...
n=1
called alternating series. Theorem: If an alternating series, either ∞
∞
n=1
n=1
∞
∞
n=1
n=1
∑ bn = ∑ ( −1) an = −a1 + a2 − a3 + a4 − +... or ∑ cn = ∑ ( −1)
n
n−1
an = a1 − a2 + a3 − a4 + −... , where an > 0 satisfy 1) bn+1 ≤ bn and 2) lim bn = 0 . Then the series are converges. n→∞