CK-12-Calculus

8.1. Sequences http://www.ck12.org The figure above shows the graph of the sequence {ln(n) n } . Notice that fromNon, the terms ...

http://www.ck12.org Chapter 8. Infinite Series Example 7 The sequence{ln(n)}grows without bound asnapproaches infinity. Note tha ...

8.1. Sequences http://www.ck12.org Rules, Sandwich/Squeeze Properties of function limits are also used with limits of sequences. ...

http://www.ck12.org Chapter 8. Infinite Series n→lim+∞ ( 11 n − 8 n^2 ) =n→lim+∞^11 n −nlim→+∞n^82 Applying the rule for the dif ...

8.1. Sequences http://www.ck12.org Thus, limn→+∞ 0 ≤limn→+∞^8 nn!≤limn→+∞(^8 n) ( 87 7! ) . By using the Rules Theorem, we have ...

http://www.ck12.org Chapter 8. Infinite Series Yn(x) =y 0 +∫xx 0 f(t,Yn− 1 (t))dtforn≥ 0 The sequence of approximations converge ...

8.1. Sequences http://www.ck12.org Y 3 (x) = 1 + ∫x − 1 t ( 5 8 + t^2 4 + t^4 8 ) dt Y 3 (x) = 1 + ( 5 t 2 16 + t^4 16 + t^6 48 ...

http://www.ck12.org Chapter 8. Infinite Series 9. { 6 n 2 en } Let{an}be a sequence such that limn→+∞|an|=0. Show that limn→+∞a ...

8.2. Infinite Series http://www.ck12.org 8.2 Infinite Series Learning Objectives Demonstrate an understanding of series and the ...

http://www.ck12.org Chapter 8. Infinite Series u 1 +u 2 +u 3 +u 4 +.... A shorthand notation for an infinite series is to use si ...

8.2. Infinite Series http://www.ck12.org Convergence/Divergence of Series Let∑∞k= 1 ukbe an infinite series and let{sn}be the se ...

http://www.ck12.org Chapter 8. Infinite Series sn=^109 ( 1 − 101 n) Now we find the limit of both sides: n→lim+∞sn=nlim→∞^109 ( ...

8.2. Infinite Series http://www.ck12.org Example 5 Determine if the series 7+^78 + 872 + 873 +...+ 8 i^7 − 1 +...converges or di ...

http://www.ck12.org Chapter 8. Infinite Series Sn= n k∑= 1 ( 2 k− 2 k+ 1 ) = ( 2 1 − 2 1 + 1 ) + ( 2 2 − 2 2 + 1 ) + ( 2 3 − 2 3 ...

8.2. Infinite Series http://www.ck12.org Determine if∑∞k= (^1) k−^83 converges or diverges. Solution Using thenth-Term Test, lim ...

http://www.ck12.org Chapter 8. Infinite Series Theorem If∑∞k= 1 ukconverges, then∑∞k= 1 uk+(u 1 +u 2 +...+um)is also convergent. ...

8.2. Infinite Series http://www.ck12.org Suppose∑∞k= 1 Ukis a convergent series and∑∞k= 1 Vkis a divergent series. Explain why∑ ...

http://www.ck12.org Chapter 8. Infinite Series 8.3 Series Without Negative Terms Learning Objectives Demonstrate an understandi ...

8.3. Series Without Negative Terms http://www.ck12.org If two fractions have the same numerator but different denominators, the ...

http://www.ck12.org Chapter 8. Infinite Series ... Intuitively, we can see that there is no bound on the sequence of partial sum ...