How to use this applet:
The three buttons in the program represent examples of the three
types of result that can be obtained from the Ratio Test. These
are a good starting point. It should be apparent from the graph
that the converging graphs are ones which ben down and diverging
graphs are those which bend up - 'neither' being in between i.e. straight.
The user may also enter series. All functions must be grouped with square brackets so the program knows how to group the terms. All series must not include spaces. Operators:
| ^ | denotes powers |
| * | denotes multiplication |
| / | denotes division |
| -,+ | denotes subtract, add |
Here are some examples which you can use to get used to the syntax and see for yourself if they converge or not...
[2]*[x]/[x]^[2]
[2]^[x]/[x+1]^[3]
[3/2]^[x]/[x]
[3]^[x]/[x+10]
[2]^[x]/[[x/2]+[x]^[1/2]]
[[x/6]+[x]^[2]]/[[3]^[x]-2*x]
[5*x]^[1/x]/[2]^[x]
Notes on the maths used in the applet:
There really is only one thing you need to know to be able to do
the ratio test and that is the formula:
r = Limit(n->infinity)[Un+1/Un]
if r=1 or r=-1 then the series does not converge or diverge
if r>1 or <-1 then the series diverges
if -1>r<1 then the series converges
Simple as that eh? Don't forget to divide
by the highest power of x before getting the limit!Other useful information:
see also:
- Sequences and Series
- Limits
