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I have

$$

T(x,t) = \sum_{n = 1}^{\infty}A_n\cos\lambda_n x\exp(-\lambda_n^2t)

$$

The eigenvalues are determined by

$$

\tan\lambda_n = \frac{1}{\lambda_n}

$$

The initial condition is $T(x,0) =1$.

For the particular case of $f(x) = 1$, numerically determine the series coefficients $A_n$ and construct a series representation for $T(x,t)$.

How do I do this?

$$

A_n = 2\int_0^1\cos\lambda_n xdx

$$