A capacitor is a small energy storage device. They come in different types. The simplest capacitor is literally just two metal plates next to each other that don’t touch. When you apply electricity to the plates, energy is stored in an electric field. When you remove the electricity, the electric field collapses and the energy that was stored gets converted back into electricity.
Capacitors have a bizillion and one uses in electronics. One of the uses is in power rectifying and filtering, which is basically, converting the AC that comes from the wall into DC that the TV can use. AC is constantly changing like a sine wave, so it goes all the way up to its peak value and back down to zero, going through a complete sine wave cycle 60 times each second in the U.S. (50 times a second in many other parts of the world). The capacitor smooths things out so that the power supply still puts out a constant voltage even when the AC sine wave is at the zero part of its cycle.
An actual TV power supply is a bit more complicated than that, but that’s the basic principle involved.
When people online are talking about capacitors going bad, they are usually referring to a specific type of capacitor that is often used in power supplies. This is called an electrolytic capacitor, and instead of being just two metal plates, it’s either one or two pieces of metal foil that is coiled up. The capacitor also has a bunch of goo slathered onto the foil to make it more efficient. That goo is called the electrolyte, which is why they are called electrolytic capacitors. If you look on a circuit board, these are the things that look like little tin cans. Inside the metal case (which is a lot like a tin can) you’ve got the foil and goo all rolled up like a fruit roll-up.
What happens with these capacitors is that sometimes the manufacturer screws up and doesn’t mix the goo up properly. There have been some famous cases where this was a result of one manufacturer stealing the goo recipe from another manufacturer, only they stole the wrong copy and the copy that they stole didn’t work very well. What usually happens is that the goo basically dries out and fails. Then the capacitor doesn’t do its job, the varying voltages don’t get smoothed out, and the TV circuitry doesn’t work properly because it’s not getting constant power any more. Sometimes when the little capacitors fail, they actually blow the entire top off of the tin can. If you look at the circuit board and see little tin cans that are swelled up and leaking goo, those capacitors have failed. Sometimes the damage isn’t so obvious, though.
Replacing the failed capacitors requires a lot of soldering skill on a modern circuit board, and when it’s not particularly obvious which capacitors have failed, the technician will often just take the “shotgun” approach and will change out a whole bunch of them, since it’s faster to do that (and hence, costs less in labor) than trying to troubleshoot and test each capacitor.
Technician time is very pricey, so in most cases you are better off just replacing the TV.
As for you being an evil murderer, it’s actually possible. You mention it being near a window that causes temperature fluctuations. Minor temperature fluctuations are probably no big deal, but if the temperature fluctuates significantly, this causes everything in the TV to heat up and cool down. Since the different components are made out of different materials, this causes thermal stresses due to different materials expanding and contracting at different rates. If the TV has any weak solder joints, the thermal expansion and contraction will cause them to open up and fail over time. So it might not be the capacitors that are failing, but might be something like this instead.
Heat also kills. A general rule of thumb is that every 10 degrees C above room temperature will cut the expected life of the device in half. So if the sun is baking it for several hours per day, not only will you have the thermal stresses causing things to break, but you’ll also have overheating that will be shortening the life of components. Heat actually makes the capacitor problem worse, as excessive heat causes the electrolyte to dry out and fail.
Another problem you might be having is bad power. If you are close to a substation, your voltage might be a bit on the high side. You might also be experiencing large power spikes during the day, as things are switched on and off of the power line. In some areas, lighting is common and tends to travel fairly long distances along power lines and through the ground, often shortening the life of devices even though the lightning never hits very close to your home. There’s all kinds of test equipment that you can hook up to your power line to monitor for these types of problems, but most of it isn’t available to the home user unless the home user happens to be an electrical engineer like me. The one thing that the home user does have available is a cheap voltmeter, which can be had for about 20 bucks. With this, you can at least see if the voltage at your house is higher than usual. The power company usually specifies 120 volts plus or minus 10 percent (maybe 5 percent in some areas, which is better). That would put the acceptable voltage range anywhere from 108 to 132 volts. If it’s on the high side of that, you might find the higher voltage causing things to wear out a little faster. You’ll notice it in light bulbs and other things too. Since you don’t mention any problems with other electronic devices, I’m guessing you probably don’t have this type of problem.
Get the TV away from the window if you can, keep the temperature constant and cooler is generally better.
Power strips aren’t necessarily surge protectors. A power strip can be just a power strip with no protection at all in it. There are several things to look for in power strips. One is the joule rating, and with this, the higher the better. A cheap one might be a couple hundred joules. A really good power strip might be 800 joules or higher. A whole house surge protector (which can be installed by an electrician) might be able to handle several thousand joules, which is definitely better.
Another thing is the max fault current that the strip can handle. Again, the higher the better. A cheapie might be a few thousand amps, where a more expensive one might be several tens of thousands of amps.
You also want to look at what is called the clamping voltage. Your power is supposed to be 120 volts. The surge protector will clamp the voltage at a particular level, which means that it won’t let the voltage exceed that level. Lower is better. A surge protector that clamps at 200 volts gives you better protection than one that clamps at 300 volts.
Ideally, you also want to get a surge protector that tells you when it fails. The better surge protectors will have a light or LED that indicates whether they are working or not. If a surge comes in, it will usually blow the protection in the surge protector (often some metal oxide varistors, which clamp down on the incoming surge and often get destroyed in the process). If the surge protector doesn’t have an indicator light, you don’t know if your surge protection was blown by an incoming surge or not, so get one that has a light.