From a fixed support O, hangs an elastic string of negligible mass and natural length L. Masses n1 and M2 attached at the lower end elongate the string to 2L. Ratio of M1/M2

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Sum of series 2[n/2] ! [n/2]!/ n! [2Co-2C2+3C2....+[-1]^n[n+1]2Cn] where n is an even positive integer. Kindly provide solution of this question and how to generalise this. How do we know that which expansion will give this?