Newton-Raphson method is used to find the roots/zero of real valued function.
I am not able to understand the correctness of the algorithm.
$$x_{n+1} = x_n - \frac{f(x_n)}{f'(x_n)}$$
my question is why the equation guarantees that it will eventually find the root of function after multiple iterations.
Here's A DIY approach. Grab a pencil and paper
(0) Draw a curve intersecting the $x$-axis at some point.
(1) Mark a point $x_o$ on the $x$-axis at some initial guess for $x$. Set $k=0$
(2) Draw a vertical line from $x_k$ on the $x$-axis till it meets the curve.
(3) From this point on the curve draw a tangent to the curve, till it intersects the $x$-axis.
(4) Set $k=k+1$ and Call this new point on the $x$-axis, $x_k$ (This is your updated value) .
Repeat Steps (2) -- (4) till convergence
It is ONLY guaranteed to work if $f(x)$ has a continuous first derivative and no saddle point (zero slope with $f(x)\ne 0$) between your initial guess and the root closest to the initial guess, if it exists.