Sequences

1. Sequences • A sequence is a list of numbers in a definite order: a1, a2,L,an,L or {an} or • Suppose n1<n2<L<nk<L, then we call a subsequence of {an}, denoted by • Some sequences can be defined by giving a formula for the general n-th term. • A sequence can be thought as a function f(n)=an, with domain N={1,2,L}

2. Example • Ex. Find a formula for the general term of the sequence • Sol. • Ex. The Fibonacci sequence is defined recursively by The first few terms are

3. Example • Ex. Find a formula for the general term of the Fibonacci sequence. • Sol. Assume that On comparing coefficients, we have Solving the quadratic equation, we get Since is a geometric sequence, we find

4. Example • This gives or, Therefore, • Question: find a formula for the general term of

5. Limit of a sequence • Definition A sequence {an} has the limitL and we write if 8e>0, 9N >0 such that |an-L|<e whenever n >N. If the limit of a sequence exists, we say the sequence converges, otherwise we say it diverges. • A sequence is a special function, so all the properties for function limits are also true for sequence limits.

6. Function limit and sequence limit • The following theorem is obvious: Theorem If and an=f(n), then

7. Properties • The Squeeze Theorem holds also true: If an·bn·cn for n¸n0 and then • Ex. Find the limit of the sequence an=n!/nn. Sol. Since 0<an·1/n, by the Squeeze Theorem,

8. Example Ex. Discuss the convergence an=rn, r2(-1,+1). Sol. (i)When |r|>1, |r|n increasingly goes to infinity. (ii)When |r|<1, |r|n decreasingly goes to zero. (iii)When r=1, an1, so the limit is 1. (iv)When r=-1, an oscillates between 1 and –1 infinitely often, so it diverges. To summarize, {rn} is convergent if –1<r·1 and divergent for all other values of r.

9. Example Ex. Find the limit Sol.

10. Monotonicity and boundedness • Definition A sequence is called increasing if for all It is called decreasing if for all It is called monotonic if it is either increasing or decreasing. • Definition A sequence is bounded above if there is a number M such that for all It is bounded below if there is a number m such that for all

11. Monotonic sequence theorem Theorem Every bounded, monotonic sequence is convergent. In particular, an increasing sequence that is bounded above is convergent; a decreasing sequence that is bounded below is convergent. Ex prove that {an} is convergent. Sol. It is easy to see and That is, {an} is increasing and bounded above, and thus converges.

12. Example Ex.Suppose {an} is defined by the recursive relationship Find the limit of {an}. Sol. so an>1 for all n. so {an} is decreasing. Let Using the recursive relationship, we have Solving the equation for L, we get L=1 or L=0. Since an>1, we eliminate L=0. Therefore

13. Example {an} is defined by Show that {an} is convergent and find its limit. Sol. We can easily prove an>1 for all n by induction. Then we have an+1<an. So {an} is decreasing and bounded below. Let Taking limits on both sides of the recurrence equality, we obtain Solving the equation for L, we get Since an>1, we discard L=-1 to get L=1.

14. Series • If is a sequence, then is called an infinite series (or just a series) and is denoted by • We call the n-th partial sum of the series. • The partial sums form a new sequence If it converges, then the series is called convergent and the number is calledthe sum of the series.We denote

15. Example • Ex. Discuss the convergence of the geometric series • Sol. If r=1, then so the series diverges. If then so the series converges when and diverges otherwise. To summarize, the geometric series is convergent if and only if and the sum is

16. Example • Ex. Is the series convergent or divergent? • Sol. Since it is a series with thus is divergent. • Ex. Find the sum of the series • Sol.

17. Example • Ex. Show that the harmonic series is divergent? • Sol.

18. Necessary condition for convergence • Theorem. If the series is convergent, then • Proof. • The test for divergence If does not exist or if then the series is divergent. • Remark. is only a necessary condition, but not sufficient.

19. Properties • Theorem If and are convergent, then • Question: converges, diverges, then both and diverges, then • Remark A finite number of terms doesn’t affect the convergence or divergence of a series.

20. Convergence test for positive series • Theorem Suppose is a series with positive terms, then it converges if and only if the sequence of the partial sum is bounded, i.e., there exists a constant M such that for all n. • Proof. Note that for a positive series, the sequence of the partial sum is always increasing. So the theorem follows immediately from the monotonic sequence theorem.

21. The integral test • Theorem Suppose f is a continuous, positive, decreasing function on and let Then the series is convergent if and only if the improper integral is convergent. • Ex. For what values of p is the series convergent? • Sol. The p-series is convergent if p>1 and divergent if

22. Example • Ex. Determine whether the series converges or diverges. • Sol. The improper integral So theseries diverges.

23. Question • Ex. Determine whether the series converges or diverges. • Sol.

24. Homework 23 • Section 11.1: 22, 35, 36, 51, 61, 62 • Section 11.2: 23, 44 • Section 11.3: 7, 20, 24