Proving that a recursive sequence converges

Question

The sequence is defined as $ x_{n} = \sum_{k=1}^{n} \frac{1}{3^{k}} $

I have re-written the sequence like so:

$x_{1} = \frac{1}{3} $ and $ x_{n} = \frac{1}{3^{n}} + x_{n-1}$

Now it's easier to work with it recursively. I believe I have proved that it increases by induction. I have proposed an upper bound of 2, but can't seem to show through induction that $ \forall n $ we get $ x_{n} < 2 $

Can someone how this is done. I haven't huge practice in working with recursive sequences. Thanks a lot !

Answer 1

It's very hard to prove an inequality like this inductively. It's much easier to write down a couple of sequence elements, guess how $x_n$ can be written in a closed expression and then prove equality to this closed expression inductively:

$x_1=\frac13$, $x_2=\frac49$, $x_3=\frac{13}{27}$, $x_4=\frac{40}{81}$, $x_5=\frac{121}{243}$, $\ldots$

If you won't (or can't) use the formula for geometric sums that's the way to go.