Prove if the positive series ∑ n=1 to ∞ a_n, ∑ n=1 to ∞ b_n both converge then ∑ n=1 to ∞ a_n*b_n converges...?

Question

also: 2) If ∑ n=1 to ∞ a_n is a positive series that converges and m is a natural number, prove that ∑ n=1 to ∞ (a_n)^m also converges.

Thank You!

Answer 1

for the first question...

since the b series converges, there must be a number k such that b_n<1 for all n>=k

look at the sum from k to infinity of a_n*b_n. a_n*b_n < a_n since b_n<1.

since a_n from 1 to infinity converges, a_n from k to infinity certainly converges. the series a_n*b_n is bounded by this series, so it too must converge. now, adding on the finite quantity a_1*b_1+a_2*b_2+...+a_(k-1)*b_(k-1) will not make it diverge since it's just a finite quantity



for the second question...

again, there must be a threshold past which a_n is always < 1. looking past this threshold, a_n^m is always less than a_n, so this series is bounded by the convergent series a_n, just like in the first part

Answer 2

* Binomial coefficient (redirect from N choose k)

coefficient of the x k term in the polynomial expansion of the binomial power (1 + x) n. ... In 1852, Kummer proved that if m and n are ...



* Arithmetic function (section χ(n) - characters)

(n) defined on the set of natural number s (i.e. positive integer s) that " ... defined by the corresponding Dirichlet series (where it converges : ...



* Fibonacci number (redirect from Fibonacci series)

(1-\varphi)^n(-1/\varphi)^n are geometric series (for n 1, 2, 3, ...) ... F_a,b(n+1) & a\varphi^ n+1+b(1-\varphi)^n+1 \ &a(\varphi^ n+\varphi^ ...



* Determinant (section n-by-n matrices)

For a fixed nonnegative integer n, there is a unique determinant ... n)If B results from A by interchanging two rows or columns, then det ...



* Series (mathematics)

numbers a n , a series is informally the result of adding all those terms together ... series in the usual sense converges and has the same sum,:\ ...



* Fourier series (section Fourier series on a general interval [a, b] )

In mathematics , a Fourier series decomposes a periodic function or periodic ... b_n & \frac 1\pi\int_-\pi^\pi x \sin(nx)\, dx -\frac 2 n\ ...



* Riemann series theorem

in a permutation so that the series converges to any given value, or even diverges . ... conditionally if the series \sum_n 1^\infty a_n ...



* Generalized continued fraction (section Roots of positive numbers)

x_0 \frac A_0 B_0 b_0, \qquad x_1 \frac A_1 B_1 \frac b_1b_0+a_1 b_1 ... begin align\boldsymbol\Tau_\boldsymbol n(z)& \frac(b_n+z)A_n-1 + a_nA_ ...



* Central limit theorem (section Lacunary trigonometric series)

tail distributions decreasing as 1/ | x | ^\alpha-1 where 1 (and ... 0 then S_n / (\sigma \sqrt n) converges in distribution to \mathrm N(0,1). ...



* Universal Serial Bus (redirect from U.S.B.)

a "bit stuff violation" if the six bits before it in the CRC are '1's. ... but SuperSpeed standard B plugs will not fit into legacy standard B ...



* Meijer G-function

If the integral converges when evaluated along the second path ... b_h \prod_j m+1^q \Gamma(1+b_h - b_j) \prod_j n+1^p\Gamma(a_j - b_h) \times:\ ...



* Floor and ceiling functions (section Series expansions)

(n-1)m+x n \right \rfloor\\ &\left\lfloor \frac x m \right \rfloor +\ ... At points of discontinuity, a Fourier series converges to a value that ...



* Divergent series

In mathematics , a divergent series is an infinite series that is not convergent , ... If a series converges, the individual terms of the ...



* Continued fraction (section Theorem 1)

any infinite continued fraction converges to an irrational number , ... This is because if a\ is an integer then if x then x 0+1/(a+1/b)\ and ...



* Methods of computing square roots (section Taylor series)

this method which converges to +3 in the reals, but to -3 in the 2-adics. ... 000 is 141, since 141 2 19881 0 and converges best for S \approx 1. ...



* Holomorphic functional calculus (section Neumann series)

Since the MacLaurin series converges everywhere, the above series ... This series, the Neumann series , converges to (z − T)−1 if :\ ...



* Stirling's approximation (redirect from Stirling series)

The next term in the o(n) is 1 | 2 ln(2π n); a more precise variant ... n \ln n - n + 1 + \sum_k 2^m \frac B_k (-1)^k k(k-1) \left( \frac 1 n^ ...



* Hilbert space

a series of vectors that converges absolutely also converges to some ... function means (1) that it is ... x and y must be positive, and (3) that ...



* Cauchy sequence

such element smaller than any preassigned, necessarily positive, value. ... all such sequences are known to converge to a limit ), the criterion ...



* Riemann zeta function (redirect from Series of reciprocal powers)

of complex argument s that analytically continues the sum of the infinite series : \ ... function at even positive integers were computed by Euler. ...

Answer 3

i think of that the sum is irrational. enable, a/b = a million/ok^p(a million) + a million/ok^p(2) + ...+a million/ok^p(n) + a million/ok^p(n+a million) +... for integers a and b. Multiply the two facets by skill of bk^p(n) and check out the dimensions of S = b/(ok^(p(n+a million)-p(n)) + ... the place the adaptation p(n+a million)-p(n) exceeds b for n>c whilst n is chosen sufficiently massive. This forces S