CK-12-Calculus

8.4. Series With Odd or Even Negative Terms http://www.ck12.org Determine if the series∑k∞= 1 (− 2 k^1 +)k+ 11 converges absolut ...

http://www.ck12.org Chapter 8. Infinite Series The series∑∞k= 1 (−^1 k)!k+^1 converges according to the Alternating Series Test ...

8.5. Ratio Test, Root Test, and Summary of Tests http://www.ck12.org 8.5 Ratio Test, Root Test, and Summary of Tests Ratio Test ...

http://www.ck12.org Chapter 8. Infinite Series We see the limitation of the Ratio Test is when lim ∣∣ ∣ana+n^1 ∣∣ ∣does not exis ...

8.5. Ratio Test, Root Test, and Summary of Tests http://www.ck12.org 1.∑∞n= 1 ( n n^2 + 1 )n 2.∑∞n= 1 ((− 1 )n(lnn) n )n 3.∑∞n= ...

http://www.ck12.org Chapter 8. Infinite Series Example 4Convergence of∑(cn)nis determined with limn→∞|cn|by the Root Test. For e ...

8.5. Ratio Test, Root Test, and Summary of Tests http://www.ck12.org MEDIA Click image to the left for use the URL below. URL: h ...

http://www.ck12.org Chapter 8. Infinite Series 8.6 Power Series Power Series and Convergence Definition (Power Series) APower Se ...

8.6. Power Series http://www.ck12.org Discuss the convergence of the series∑∞n= 02 nx^2 n. Hint: Apply a combination of tests i ...

http://www.ck12.org Chapter 8. Infinite Series Term-by-Term Differentiation of Power Series The goal of the next 3 sections is t ...

8.6. Power Series http://www.ck12.org Term-by-Term Integration of Power Series Example 1 Find a power series forh(x) =tan−^1 xan ...

http://www.ck12.org Chapter 8. Infinite Series 1 ( 1 −x)( 1 − 2 x)= (^1 +x+x (^2) +...)( 1 + 2 x+ 4 x (^2) +...) = 1 + 3 x+ 7 x^ ...

8.7. Taylor and Maclaurin Series http://www.ck12.org 8.7 Taylor and Maclaurin Series Taylor and Maclaurin Polynomials We know th ...

http://www.ck12.org Chapter 8. Infinite Series f(x) =lnxatx= 1 ,n= 4 f(x) = 1 +x+x^2 +x^3 +x^4 atx=− 1 ,n= 4 Taylor and Maclau ...

8.7. Taylor and Maclaurin Series http://www.ck12.org Then f 1 ′(x) =    ( 2 x^3 )·e−x (^12) x 6 = 0 0 x= 0 It can be verifi ...

http://www.ck12.org Chapter 8. Infinite Series f′(x) =^12 ( 1 +x)−^12 ,f′′(x) =^12 ( −^12 ) ( 1 +x)−^32 =−^14 ( 1 +x)−^32 , f′′′ ...

8.7. Taylor and Maclaurin Series http://www.ck12.org by dividing into sum of odd and even indices. So cosx+isinx=∑∞m= 0 (− 1 )m( ...

http://www.ck12.org Chapter 8. Infinite Series So if|x|< 1 ,√ 1 +x=∑∞k= 1 (k^21 )xk= 1 +∑∞k= 1 ( 2 − 21 k−)k 1 −k^1 !((^2 k−k ...

8.7. Taylor and Maclaurin Series http://www.ck12.org This pattern repeats and limn→∞Rn(x) =0 can be checked as in the casex 0 =0 ...

http://www.ck12.org Chapter 8. Infinite Series ex x= e·eu 1 +u=e· 1 1 +u ∞ n∑= 0 (u)n n! =e( 1 −u+u^2 −u^3 +...) ( 1 +u+u 2 2!+ ...