you could use a ne555 to drive an opamp

By far, the 555 timer is the easiest way to go. As far as what you mean by "swing" you may have to do a little more work because the 555 produces a square wave output. Still, with some simple resistors and a capacitor, you can pretty much get what you want.

If you are trying to keep the off time equal to the on time (duty cycle) you may have to be a little more particular as to the value of fixed resistors and potentiometer you choose. There are hundreds of examples and uses for the 555 on the internet.

editing....

I just read your last question and now understand a little better. Will reply again soon.

Try this one: http://www.dprg.org/tutorials/2005-11a/index.html

By changing R1 from a common resistor to both charge and discharge cycles, into a fixed for one and variable resistor for the other; you should be able to achieve what you are trying.

You will need some additional circuitry to limit it between 1-3V.

P.S. The generic term for this 555 circuit is an "Astable Multivibrator". Doing a search for that term will bring even more circuits.

The key to using the 555 is to consider the split resistor combination for the two halves of the cycle. In order to achieve a particular duty cycle (10% would mean on for 10% of the cycle length, off for 90%) you can choose the combination of resistor and capacitor that meets your requirement.

As for the 1 volt to 3 volt signal, you can use a voltage divider with two forward biased diodes to make sure you always have at least 1 volt on the minimum side and the divider output for the other or a voltage follower transistor arrangement limited to a supply of 3 volts, if necessary. The key is in the terms used above. A voltage follower is just a single transistor with the load connected to the emitter (also known as a common collector) arrangement.

A Sin wave with 1V amplitude and a 2V DC offset swings like that quite repetitively.

You could try and revive the old project i built in the 1970s

Do a search on the web "ETI Sweep generator forum" and look for a vintage electronics site. The sine wave generator may well be fine as it is however the timing capacitor in the ramp generator may need to be lowered a bit. The transistor array will have to be made with discrete transistors as they are in short supply now. The scope sync output is the one you would use to clock the servo. If you follow this route you would have to request a library password from the forum moderator to get the data. You just need an isolated DC supply to lift it a couple of volts to get what you require. This will help you learn what is inside a modern signal generator chip if anything else.

C8, C11 and C12 spring to mind.