How do you evaluate an infinite series?

Question

Ezra Adams · Accepted Answer

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There are different types of series, to what use different methods of evaluating For example a converging geometric series: #a+ ar + ar^2 + ar^3 + ... + ar^k = sum_(n=1) ^(k) ar^(n-1) # where #sum_(n=1) ^(k) ar^(n-1) = (a(1-r^k)) / (1-r) # Assuming #|r| < 1 # we can let #k to oo # for infinite series to be evaluated ... #lim_(k to oo ) sum_(n=1) ^(k) ar^(n-1) = lim_(k to oo ) ( a(1-r^k) )/(1-k) # as #k to oo # , #r^k to 0 # as #|r|< 1 # #=> sum_(n=1) ^(oo) ar^(n-1) = a/(1-r) # but there are other series what can be approached with tricks! Take # 1/6 + 1/12 + 1/20 + 1/30 + ... # After consideration we can recognise this is the same as... # (1/2 - 1/3 ) + ( 1/3 - 1/4) + (1/4 - 1/5 ) + ... # # (1/2 cancel(- 1/3) ) + (cancel( 1/3) cancel(- 1/4)) + (cancel(1/4) cancel(- 1/5) ) + ... # #=1/2 # There are also other infinite series that you can remember, and may be able to prove, a like: #e^x = 1 + x + x^2 /(2!) + x^3 / (3!) + ... = sum_(n=0) ^oo x^n / (n!) # There are many others, where there insist one set way of computing infinite series, there are many!

Ezra Adams · Answer

undefined To evaluate an infinite series, you typically use various convergence tests to determine whether the series converges (has a finite sum) or diverges (has an infinite sum). Some common convergence tests include the comparison test, the ratio test, the root test, the integral test, and the alternating series test. Once you have determined that the series converges, you can use appropriate techniques such as geometric series summation, telescoping series summation, or other methods to find the sum of the series. If the series diverges, it means it does not have a finite sum.