Question 473442
Presuming you want


*[tex \LARGE \sum_{n=0}^{\infty} \frac{2^{n+1}3^n + 3^{n+1}5^n + 5^{n+1}7^n}{6^n15^n35^n}],


this is equal to


*[tex \LARGE \sum_{n=0}^{\infty} \frac{2^{n+1}3^n + 3^{n+1}5^n + 5^{n+1}7^n}{2^n3^n3^n5^n5^n7^n}]


*[tex \LARGE = \sum_{n=0}^{\infty} \frac{2(2^n)(3^n)}{2^n3^n3^n5^n5^n7^n} + \frac{3(3^n)(5^n)}{2^n3^n3^n5^n5^n7^n} + \frac{5(5^n)(7^n)}{2^n3^n3^n5^n5^n7^n}]


*[tex \LARGE = \sum_{n=0}^{\infty} \frac{2}{525^n} + \frac{3}{140^n} + \frac{5}{90^n}]


This can be split into three sums, all of them convergent.


*[tex \LARGE = 2(\frac{525}{524}) + 3(\frac{141}{140}) + 5(\frac{91}{90})]


And you can simplify from there.