linear algebra – Find largest eigenvalue of specific nonnegative matrix

I look for the largest eigenvalue of the following matrix (or at least a small upper bound).
The only thing I know is that the eigenvalue is smaller than 1 and converges to 1 with growing n.

In general, it is very hard to compute the characteristic polynomial to calculate the eigenvalue and that’s why I hope for an easier way.

Has anyone some ideas?

$A = begin{bmatrix}
frac{1}{2-frac{1}{n+1}}& frac{1}{2} & 0 & 0 & dots & 0 \
frac{1}{2} & 0 & frac{1}{2} & 0 & dots & 0 \
0 & frac{1}{2} & 0 & frac{1}{2} & dots & 0 \
vdots & vdots & vdots & vdots & vdots & 0 \
0 & 0 & 0 &0 & frac{1}{2} & 0
end{bmatrix}$