We have this thing as equal to the limit comes out to be 0, so the series is convergent, so this implies a radius of convergence is infinity and therefore the interval of convergence is minus infinity to infinity hope, you'll find the video useful. This is further equal to now on putting the limit.
#Calcpad interval notation plus#
This is 6 by 2 n plus 2 and 2 n plus 1 point now. Simplifying gives limit and tends to infinity 16 x to the power 6. Now, using ratio test limit and tends to infinity more a n plus 1 by a n is equal to limit and tends to infinity more 4 x cube whole to the power 2 n plus 1 by 2 n plus 1 factorial by 4 x cube whole to The power 2 n by 2 n factorial, so this on simplifying, gives the value equal to limit and tends to infinity 4 square into x to the power 6 into trun factorial by 2 n plus 2 factorial, which is equal to this 1.
So we see that now a n is equal to n, is equal to 4 x, cube to the power 2 n by 2 n factorial.
Now we have to find the interval of convergence for this a n. So therefore, we have the tailor series expansion, as summation n is equal to 0 to infinity minus 1 to the power n 4 x, cubed to the power 2 n by 2 n factorial and its first 3 terms are 1 minus 8 x to the power 6 Plus 32 by 3 x to the power 12. So now, on expanding its first 3 terms are 1 minus 4 x cubed whole square by 2, factorial plus 1 by 4, factorial 4 x cubed to the power 4. Empty set emptySets Union unionSets Intersection intersectSets Lowercase Greek letter Name of the letter in lowercase, for example, alpha, beta, gamma. Submission n is equal to 0 to infinity minus 1 to the power n 4 x, cubed whole to the power 2 n by 2 n factorial. Open interval (parentheses) (a,b)Sets Half-closed interval (half-open interval) a,b) (a,b Sets or You cannot type this notation.
We will replace x instead of 4 x cubed, so we have for cos 4 x cubed. So now for f x is equal to cos 4 x cubed. We have the tailor series, as summation n is equal to 0 to infinity minus 1 to the power n x to the power 2 n by 2 n factorial. Hi now we have to write the tailor series for f x is equal to cos 4 x cubed now for cos x.
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