TITLE: Is this algebraic identity obvious? $\sum_{i=1}^n \prod_{j\neq i} {\lambda_j\over \lambda_j-\lambda_i}=1$
QUESTION [24 upvotes]: If $\lambda_1,\dots,\lambda_n$ are distinct positive real numbers, then 
$$\sum_{i=1}^n \prod_{j\neq i} {\lambda_j\over \lambda_j-\lambda_i}=1.$$
This identity follows from a probability calculation that you can find at the 
top of page 311 in the 10th edition of Introduction to Probability Models by Sheldon Ross. 
Is there a slick or obvious explanation for this identity? 
This  question is sort of similar to my previous problem; clearly algebra is not my strong suit!

REPLY [31 votes]: It's the Lagrange interpolation polynomial for the constant function $1$ evaluated at $0$:
$$\sum_{i=1}^n 1 \prod_{j\neq i} {{\lambda_j-0}\over \lambda_j-\lambda_i}=1$$
In general, you have that the polynomial below interpolates the data points $(\lambda_i,y_i$):
$$\sum_{i=1}^n y_i \prod_{j\neq i} {{\lambda_j-x}\over \lambda_j-\lambda_i}$$
See http://en.wikipedia.org/wiki/Lagrange_polynomial