next up previous
Next: Series: direct computations Up: Review for midterm no. Previous: Review for midterm no.

Sequences

There can be three types of problems:

(a)
Compute the limit of a sequence directly.
(a')
Compute the limit of a sequence, using squeeze theorem or similar.
(b)
Show that a sequence converges. (Use the monotonic sequence theorem!)
(c)
Compute the limit of a sequence indirectly, assuming you know it converges.
In type (a) you still have to watch out whether the series converges at all.
(a)
$ a_n = \frac{3n + 1}{\sqrt{n^2 + 1}}$
(a')
$ b_n = \frac{\cos n}{\sqrt{n}}$
(a)
$ a_n = (-1)^n \frac{2n}{n+1}$
(a)
$ \frac 11, \frac 22, \frac 34, \frac 48, \dots$
(b)
$ a_n = \left(1- \frac 12\right) \left(1 - \frac 13\right) \cdots$
(c)
$ u_1 = 0, u_{n+1} = \sqrt{2 + u_n}$



Arend Bayer 2006-12-15