Why output of square wave signal after bypass capacitor is a triangular wave?
The Fourier series for a triangle wave is Odd harmonics shift in phase by 90 deg from a square wave with 12 dB/ odd octave more attenuation.



and the square wave 



Both have only odd harmonics but differ in the slope of the peak value for each harmonic. However triangle harmonics are much smaller. As n increases, the amplitude reduces by 1/n², whereas a square wave reduces by 1/n.  Triangle waves harmonics also alternate phase (+/-sin) with increasing n.

To simplify my explanation, the capacitive load on a 50Ω gen. gives a frequency response or Transfer Function of;

 , which you know gives the exponential time response to a medium square wave  where f is near 1/RC .

But for high frequency where RCs>>1 so the transfer fcn reduces to an integrator Hc(s)=1/RCs transfer function. 

Applying this filter to the Fourier series of the square wave, its 1/n harmonic attenuation becomes 1/n² slope on harmonics of the triangle wave.  Similarily the triangle wave source when filtered, its harmonic attenuation slope of 1/n²  becomes 1/n³.
On a scope all you would see is a sine wave, but on a spectrum analyzer log scale you would see the 1/n³ slope of all harmonics ( i.e. 3rd order slope )

side comments added
I believe there is value in the time you spend in the lab to match theory with practise.   When it does not match, look at for a better equivalent circuit then verify your assumptions.  If you have Java you can play with this programmable signal generator .  Have fun !    Spend more time in the lab validating what you learn and bring the lab to your desktop then expand your horizons.

http://www.falstad.com/fourier/  &lt;

use the mouse pointer and left or right click...drag and adjust


Change the phase of the Fourier series and see the effect on a triangle wave
Add a spurious resonance in the amplitude of one harmonics, see the waveform
add a glitch, change the shape anyway you want. (arbitrary waveform)
add a LPF filter, change the frequency, slide the number of terms in the spectrum, see effect


When Integrate a step input and Dump, you get a Sawtooth waveform

Here the mouse is hovering over the fundamental of the Fourier spectrum and the fundamental sinewave amplitude and phase are shown in yellow. Meanwhile I boosted a harmonic to similate a resonance on the sawtooth.

The combinations are "only limited by your imagination"