Calculate the Fourier series approximation up to the 5th harmonics?

First, compute the Fourier coefficients.

a₀ = (1/π) ∫(x = -π to π) f(t) dt
.....= (1/π) [0 + ∫(x = 0 to π) π dt]
.....= π.

For n > 0:
an = (1/π) ∫(x = -π to π) f(t) cos(nπt/π) dt
.....= (1/π) [0 + ∫(x = 0 to π) π cos(nt) dt]
.....= (1/n) sin(nt) {for x = 0 to π}
.....= 0.

bn = (1/π) ∫(x = -π to π) f(t) sin(nπt/π) dt
.....= (1/π) [0 + ∫(x = 0 to π) π sin(nt) dt]
.....= (-1/n) cos(nt) {for x = 0 to π}
.....= ((-1)^(n+1) + 1)/n
.....= 2/n when n is odd and 0 when n is even.

Therefore,
f(t) ~ (1/2) a₀ + Σ(n = 1 to ∞) [an cos(nt) + bn sin(nt)]
......= (1/2) π + Σ(n = 1 to ∞) [0 cos(nt) + bn sin(nt)]
......= π/2 + Σ(n = 1 to ∞) bn sin(nt)
......= π/2 + [(2/1) sin(1t) + 0 + (2/3) sin(3t) + 0 + (2/5) sin(5t) + ...)
......= π/2 + 2 sin t + (2/3) sin(3t) + (2/5) sin(5t) + ... .

The rest is straightforward; let t = 1.15, and truncate the series after the sin(5t) term for the approximation.

I hope this helps!