سریها و سریهای توانی
Series & Power Series
آزمونهای همگرایی، شعاع و بازهٔ همگرایی.
ریاضی عمومی کنکور ارشد علوم کامپیوتر
کارتها · 6
A geometric series with r < 1 converges to (first term) / (1 - r).
The radius of a power series is 1/limsup c_n ^(1/n) — the root test that always works, even when the ratio test fails.
An alternating series converges if its terms shrink monotonically to 0; combine with absolute convergence to label it absolutely, conditionally, or divergent.
If a series' terms are eventually trapped between 0 and a constant multiple of a known convergent series, it converges too — the engine behind reasoning about square-summable sequences.
To find [x^n] of an integral with a variable upper limit, expand the integrand as a power series, then bump each exponent up by one and divide: the x^n coefficient is c_(n-1)/n.
Sum series like sum n/(2n+1)! by writing the numerator as a combination of the factorial's argument, so each term collapses onto a known e, sinh, or cosh series.