00:01
Here, we can start off by writing out the power series for cosine of x.
00:06
So, cosine of x is equal to the sum that starts at n equals 0 to infinity of negative 1 to the power of n times x to the power of 2n divided by 2 times n factorial.
00:18
In parentheses, of course, and we can write out the terms, say for example, if n equals 0, then we get 1, minus x squared over 2 factorial, plus x to the 4th, powered over 4 factorial.
00:29
And then minus xxxxx over six vectoral plus that that that that now we want to estimate we want we want to estimate a cosine of 0 .22 so that the error so that the error is less than 10 to the power of negative 6 so what does it mean well in this case by the alternating series estimation theorem which pretty much summarized says that the absolute value of the remainder at x is always going to be lessened which is defined by the way as the absolute value of the actual sum minus the approximation that this is going to be lessen or equal to bs of n plus 1 of capital n plus 1 where b's of n in this case is x to the power of 2n divided by 2n factorial in other words what we want is we want these this will imply that when x is equal to 0 .22 where we're going we want 0 .22 to the power of 2 times n plus 1 factorial divided by 2 times n plus 1 factorial to be less than 10 to the power of negative 6.
02:20
Well, consider these expression here on the left side.
02:25
When n equals 1, this is approximately equal to, well, we get a value, so when n equals 1, we get an estimate to have a rough idea of what this is, this would be roughly 0 .000 -0 -976...