Machin series approximation to 100 digits

Questions
1

I am trying to solve for the term of the Machin series which gives the approximation of pi correctly to 100 digits. I have tried:

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1e.png1610Ɨ1124 13.2 KB

Can anyone tell me what Iā€™m doing wrong and how i can solve it?
Thanks in advance :slight_smile:

2

The first summand has much larger absolute value than the second, so we could solve a simpler, approximate equation

In[4]:= NSolve[16 (1/5)^(2 k + 1)/(2 k + 1) == 10^-102, k, Reals]

Out[4]= {{k -> 71.7807}}

Then compute the terms with k = 71 and k = 72 and see if they are acceptable.