mmtheorems73 - Metamath Proof Explorer

Home	Metamath Proof Explorer	< Previous Next >
		Browser slow? Try the Unicode version.

Jump to page: Contents + 1 1-100 2 101-200 3 201-300 4 301-400 5 401-500 6 501-600 7 601-700 8 701-800 9 801-900 10 901-1000 11 1001-1100 12 1101-1200 13 1201-1300 14 1301-1400 15 1401-1500 16 1501-1600 17 1601-1700 18 1701-1800 19 1801-1900 20 1901-2000 21 2001-2100 22 2101-2200 23 2201-2300 24 2301-2400 25 2401-2500 26 2501-2600 27 2601-2700 28 2701-2800 29 2801-2900 30 2901-3000 31 3001-3100 32 3101-3200 33 3201-3300 34 3301-3400 35 3401-3500 36 3501-3600 37 3601-3700 38 3701-3800 39 3801-3900 40 3901-4000 41 4001-4100 42 4101-4200 43 4201-4300 44 4301-4400 45 4401-4500 46 4501-4600 47 4601-4700 48 4701-4800 49 4801-4900 50 4901-5000 51 5001-5100 52 5101-5200 53 5201-5300 54 5301-5400 55 5401-5500 56 5501-5600 57 5601-5700 58 5701-5800 59 5801-5900 60 5901-6000 61 6001-6100 62 6101-6200 63 6201-6300 64 6301-6400 65 6401-6500 66 6501-6600 67 6601-6700 68 6701-6800 69 6801-6900 70 6901-7000 71 7001-7100 72 7101-7200 73 7201-7300 74 7301-7400 75 7401-7500 76 7501-7600 77 7601-7700 78 7701-7800 79 7801-7900 80 7901-8000 81 8001-8100 82 8101-8200 83 8201-8300 84 8301-8400 85 8401-8500 86 8501-8600 87 8601-8700 88 8701-8800 89 8801-8900 90 8901-9000 91 9001-9100 92 9101-9200 93 9201-9300 94 9301-9400 95 9401-9500 96 9501-9600 97 9601-9700 98 9701-9800 99 9801-9900 100 9901-10000 101 10001-10100 102 10101-10200 103 10201-10300 104 10301-10400 105 10401-10500 106 10501-10600 107 10601-10700 108 10701-10752

Color key:

Metamath Proof Explorer
(1-8782)

Hilbert Space Explorer
(8783-10363)

User Sandboxes
(10364-10752)

**Statement List for Metamath Proof Explorer -** 7201-7300 - Page 73 of 108
Type	Label	Description
Statement

Theorem	isumclim4t 7201	An infinite sum equals the value its series converges to.
$/\$ $/\$

Theorem	isumclim5t 7202	The sequence of partial finite sums of a converging infinite series converge to the infinite sum of the series. Note that must not occur in .
$/\$ $/\$

Theorem	isumnul 7203	The sum of a non-convergent infinite series evaluates to the empty set. (Contributed by Paul Chapman, 04-Nov-2007.)
$/\$

Theorem	isumshft 7204	Index shift of an infinite sum. (Contributed by Paul Chapman, 31-Oct-2007.)


Theorem	isumshft2 7205	Index shift of an infinite sum. (Contributed by Paul Chapman, 31-Oct-2007.)


Theorem	isum1p 7206	The infinite sum of a converging infinite series equals the first term plus the infinite sum of the rest of it.
$/\$ $/\$

Theorem	isumnn0nn 7207	Sum from 0 to infinity in terms of sum from 1 to infinity.
$/\$

Theorem	isumnn0nna 7208	Sum from 0 to infinity in terms of sum from 1 to infinity of a class term .
$/\$

Theorem	isumclt 7209	The sum of a converging infinite series is a complex number.
$/\$

Theorem	isumreclt 7210	The sum of a converging real infinite series is a real number.
$/\$ $/\$

Theorem	iserzgt0 7211	The infinite sum of positive reals is positive. (Contributed by Paul Chapman, 9-Feb-2008.)
$/\$ $/\$ $/\$

Theorem	isummulc1 7212	Distribute a constant multiplier into an infinite sum (function value version). See isummulc1a 7214 for class term version.
$/\$ $/\$ $/\$

Theorem	isummulc1ALT 7213	Older isummulc1 7212 proved without using iserzmulc1 7136.
$/\$ $/\$ $/\$

Theorem	isummulc1a 7214	Distribute a constant multiplier into an infinite sum of a class term . See isummulc1 7212 for function value version.
$/\$ $/\$ $/\$

Theorem	isumcmpi 7215	Comparison of two infinite sums. (Contributed by Paul Chapman, 13-Nov-2007.)
$/\$ $/\$ $/\$ $/\$ $/\$ $/\$

Theorem	isumsplit 7216	Split off the first terms of an infinite sum. (Contributed by Paul Chapman, 9-Feb-2007.)


Theorem	isum0split 7217	Split off the first terms of a 0-based infinite sum. (Contributed by Paul Chapman, 9-Feb-2007.)


Miscellaneous converging sequences

Theorem	reccnv 7218	The sequence of reciprocals of natural numbers converges to zero.


Theorem	infcvgaux1 7219	Auxilliary theorem for applications of infcvg 7224. Hypothesis for several supremum theorems.
$/\$ $/\$

Theorem	infcvgaux2 7220	Auxilliary theorem for applications of infcvg 7224.


Theorem	infcvglem1 7221	Lemma for infcvg 7224. Use ac6s 4766 to show the existence of a sequence with values extracted from .

Theorem	infcvglem2 7222	Lemma for infcvg 7224. Show that converges to the infimum.

Theorem	infcvglem3 7223	Lemma for infcvg 7224. Using climsqueeze 7140, show that sequence , constructed from , converges to the supremum.

Theorem	infcvg 7224	Extract a sequence that converges to the infimum of a set of reals . The sequence is built using values of a sequence that converges when its values are mapped to reals via . Equation 4 of [Kreyszig] p. 144.
$/\$

Arithmetic series

Theorem	fnsmntlem 7225	Lemma for fnsmnt 7226.

Theorem	fnsmnt 7226	Arithmetic series sum of the first non-negative integers. (Contributed by FL, 10-Dec-2006.)


Geometric series

Theorem	expcnvlem1 7227	Lemma for expcnv 7233. Convert an antecedent from a comparison with a real into comparison with a natural number.

Theorem	expcnvlem2 7228	Lemma for expcnv 7233. Compute an upper bound for exponentiation using Bernoulli's inequality bernneq 6653.

Theorem	expcnvlem3 7229	Lemma for expcnv 7233. Apply weak deduction theorem.

Theorem	expcnvlem4 7230	Lemma for expcnv 7233. Combine expcnvlem1 7227 and expcnvlem3 7229.

Theorem	expcnvlem5 7231	Lemma for expcnv 7233. Apply weak deduction theoerem.

Theorem	expcnvlem6 7232	Lemma for expcnv 7233. Add in the case of .

Theorem	expcnv 7233	A sequence of powers of a complex number with absolute value smaller than 1 converges to zero.
$/\$ $/\$