Other Useful facts a n converges to zero if and only if ja njalso converges to zero. Squeeze theorem If b n a n c n for all values of n, and limb n = limc n = L, then it implies that lima n = L. Review Sheet for Calculus 2 Sequences and Series SEQUENCES Convergence A sequence fa ngconverges if lima n exists and is nite. ![]() Join millions of learners from around the world already learning on Udemy Improved: UI, security, and stability with updated framework and libraries. In order to de ne a sequence we must give enough information to nd itsn-th term. E.g.: 1 2 3 4 ::: We represent a generic sequence asa1 a2 a3 :::,anditsn-th asa n. Give yourself a better future: 1-on-1 calculus tutoring changes lives. An in nite sequence of real numbers is an ordered unending list of real numbers. Function as a geometric series.ĭon't let anything stop you from achieving your best calculus results. The number multiplied is called the common ratio. Geometric sequences happen when you multiply numbers. The number added is called the common difference. Quiz 2: 8 questions Practice what you’ve learned, and level up on the above skills. Sequences and Series Cheat Sheet Arithmetic Sequences and Series Geometric Sequences and Series Arithmetic sequences happen when you add numbers. ![]() Important Series to Remember Series How do they look Conclusions Geometric Series. This intro-duces some hints and some ideas you may consider when choosing tests for convergence or divergence when evaluating a given series.įor any series X∞ n=0 a n, there are 3 cases: Ratio Test: Calculate lim n→∞ a n+1 a n = L if L 1, then X∞ n=0 |a n| diverges if L = 1, no conclusion can be made. Professionally curated courses developed with employer and industry specialist input.Ĭalculus II Series - Things to Consider Important: This cheat sheet is not intended to be a list of guaranteed rules to follow. An explicit formula for this arithmetic sequence is given by an a+(n1)b, n N, a recursive formula is given by a1 a and an an1+ b for n > 1. Ranked #1 in the UK for student satisfaction, student voice and academic support in 2023. But you can take some of the fear of studying calculus away by understanding its basic principles, such as derivatives and antiderivatives, integration, and solving compound functions. By its nature, calculus can be intimidating. It looks like the Nth partial sum of this series is 2N 1 2N D. AP Calculus AB Cheat Sheet Practice Exams: (We will be going over. For example, the partial sums of the Meg Ryan series 1 2 + 1 4 + 1 8 + are: 1st partial sum 1 2 2nd partial sum 1 2 + 1 4 3 4 3rd partial sum 1 2 + 1 4 1 8 7 8 and so forth. That has saved us all a lot of trouble! Thank you Leonardo.įibonacci Day is November 23rd, as it has the digits "1, 1, 2, 3" which is part of the sequence.Calculus II For Dummies. 1d Limit Representations of the Derivative - 2. "Fibonacci" was his nickname, which roughly means "Son of Bonacci".Īs well as being famous for the Fibonacci Sequence, he helped spread Hindu-Arabic Numerals (like our present numbers 0, 1, 2, 3, 4, 5, 6, 7, 8, 9) through Europe in place of Roman Numerals (I, II, III, IV, V, etc). ![]() His real name was Leonardo Pisano Bogollo, and he lived between 11 in Italy. Historyįibonacci was not the first to know about the sequence, it was known in India hundreds of years before! Which says that term "−n" is equal to (−1) n+1 times term "n", and the value (−1) n+1 neatly makes the correct +1, −1, +1, −1. In fact the sequence below zero has the same numbers as the sequence above zero, except they follow a +-+. (Prove to yourself that each number is found by adding up the two numbers before it!) Innite Sequence: hsni Limit/Convergence of a Sequence: limn 1sn L Innite Serie: (Partial sums) Sn P sn s1+ s2+ + sn+ Geometric Serie: Xn k 1.
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