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Mirrors > Home > MPE Home > Th. List > divsqrtsum2 | Structured version Visualization version GIF version |
Description: A bound on the distance of the sum Σ𝑛 ≤ 𝑥(1 / √𝑛) from its asymptotic value 2√𝑥 + 𝐿. (Contributed by Mario Carneiro, 18-May-2016.) |
Ref | Expression |
---|---|
divsqrtsum.2 | ⊢ 𝐹 = (𝑥 ∈ ℝ+ ↦ (Σ𝑛 ∈ (1...(⌊‘𝑥))(1 / (√‘𝑛)) − (2 · (√‘𝑥)))) |
divsqrsum2.1 | ⊢ (𝜑 → 𝐹 ⇝𝑟 𝐿) |
Ref | Expression |
---|---|
divsqrtsum2 | ⊢ ((𝜑 ∧ 𝐴 ∈ ℝ+) → (abs‘((𝐹‘𝐴) − 𝐿)) ≤ (1 / (√‘𝐴))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | divsqrsum2.1 | . 2 ⊢ (𝜑 → 𝐹 ⇝𝑟 𝐿) | |
2 | divsqrtsum.2 | . . . 4 ⊢ 𝐹 = (𝑥 ∈ ℝ+ ↦ (Σ𝑛 ∈ (1...(⌊‘𝑥))(1 / (√‘𝑛)) − (2 · (√‘𝑥)))) | |
3 | 2 | divsqrtsumlem 26280 | . . 3 ⊢ (𝐹:ℝ+⟶ℝ ∧ 𝐹 ∈ dom ⇝𝑟 ∧ ((𝐹 ⇝𝑟 𝐿 ∧ 𝐴 ∈ ℝ+) → (abs‘((𝐹‘𝐴) − 𝐿)) ≤ (1 / (√‘𝐴)))) |
4 | 3 | simp3i 1141 | . 2 ⊢ ((𝐹 ⇝𝑟 𝐿 ∧ 𝐴 ∈ ℝ+) → (abs‘((𝐹‘𝐴) − 𝐿)) ≤ (1 / (√‘𝐴))) |
5 | 1, 4 | sylan 580 | 1 ⊢ ((𝜑 ∧ 𝐴 ∈ ℝ+) → (abs‘((𝐹‘𝐴) − 𝐿)) ≤ (1 / (√‘𝐴))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 = wceq 1541 ∈ wcel 2106 class class class wbr 5103 ↦ cmpt 5186 dom cdm 5631 ⟶wf 6489 ‘cfv 6493 (class class class)co 7351 ℝcr 11008 1c1 11010 · cmul 11014 ≤ cle 11148 − cmin 11343 / cdiv 11770 2c2 12166 ℝ+crp 12869 ...cfz 13378 ⌊cfl 13649 √csqrt 15077 abscabs 15078 ⇝𝑟 crli 15326 Σcsu 15529 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2708 ax-rep 5240 ax-sep 5254 ax-nul 5261 ax-pow 5318 ax-pr 5382 ax-un 7664 ax-inf2 9535 ax-cnex 11065 ax-resscn 11066 ax-1cn 11067 ax-icn 11068 ax-addcl 11069 ax-addrcl 11070 ax-mulcl 11071 ax-mulrcl 11072 ax-mulcom 11073 ax-addass 11074 ax-mulass 11075 ax-distr 11076 ax-i2m1 11077 ax-1ne0 11078 ax-1rid 11079 ax-rnegex 11080 ax-rrecex 11081 ax-cnre 11082 ax-pre-lttri 11083 ax-pre-lttrn 11084 ax-pre-ltadd 11085 ax-pre-mulgt0 11086 ax-pre-sup 11087 ax-addf 11088 ax-mulf 11089 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2887 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-rmo 3351 df-reu 3352 df-rab 3406 df-v 3445 df-sbc 3738 df-csb 3854 df-dif 3911 df-un 3913 df-in 3915 df-ss 3925 df-pss 3927 df-nul 4281 df-if 4485 df-pw 4560 df-sn 4585 df-pr 4587 df-tp 4589 df-op 4591 df-uni 4864 df-int 4906 df-iun 4954 df-iin 4955 df-br 5104 df-opab 5166 df-mpt 5187 df-tr 5221 df-id 5529 df-eprel 5535 df-po 5543 df-so 5544 df-fr 5586 df-se 5587 df-we 5588 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-pred 6251 df-ord 6318 df-on 6319 df-lim 6320 df-suc 6321 df-iota 6445 df-fun 6495 df-fn 6496 df-f 6497 df-f1 6498 df-fo 6499 df-f1o 6500 df-fv 6501 df-isom 6502 df-riota 7307 df-ov 7354 df-oprab 7355 df-mpo 7356 df-of 7609 df-om 7795 df-1st 7913 df-2nd 7914 df-supp 8085 df-frecs 8204 df-wrecs 8235 df-recs 8309 df-rdg 8348 df-1o 8404 df-2o 8405 df-er 8606 df-map 8725 df-pm 8726 df-ixp 8794 df-en 8842 df-dom 8843 df-sdom 8844 df-fin 8845 df-fsupp 9264 df-fi 9305 df-sup 9336 df-inf 9337 df-oi 9404 df-card 9833 df-pnf 11149 df-mnf 11150 df-xr 11151 df-ltxr 11152 df-le 11153 df-sub 11345 df-neg 11346 df-div 11771 df-nn 12112 df-2 12174 df-3 12175 df-4 12176 df-5 12177 df-6 12178 df-7 12179 df-8 12180 df-9 12181 df-n0 12372 df-z 12458 df-dec 12577 df-uz 12722 df-q 12828 df-rp 12870 df-xneg 12987 df-xadd 12988 df-xmul 12989 df-ioo 13222 df-ioc 13223 df-ico 13224 df-icc 13225 df-fz 13379 df-fzo 13522 df-fl 13651 df-mod 13729 df-seq 13861 df-exp 13922 df-fac 14127 df-bc 14156 df-hash 14184 df-shft 14911 df-cj 14943 df-re 14944 df-im 14945 df-sqrt 15079 df-abs 15080 df-limsup 15312 df-clim 15329 df-rlim 15330 df-sum 15530 df-ef 15909 df-sin 15911 df-cos 15912 df-pi 15914 df-struct 16978 df-sets 16995 df-slot 17013 df-ndx 17025 df-base 17043 df-ress 17072 df-plusg 17105 df-mulr 17106 df-starv 17107 df-sca 17108 df-vsca 17109 df-ip 17110 df-tset 17111 df-ple 17112 df-ds 17114 df-unif 17115 df-hom 17116 df-cco 17117 df-rest 17263 df-topn 17264 df-0g 17282 df-gsum 17283 df-topgen 17284 df-pt 17285 df-prds 17288 df-xrs 17343 df-qtop 17348 df-imas 17349 df-xps 17351 df-mre 17425 df-mrc 17426 df-acs 17428 df-mgm 18456 df-sgrp 18505 df-mnd 18516 df-submnd 18561 df-mulg 18831 df-cntz 19055 df-cmn 19522 df-psmet 20740 df-xmet 20741 df-met 20742 df-bl 20743 df-mopn 20744 df-fbas 20745 df-fg 20746 df-cnfld 20749 df-top 22194 df-topon 22211 df-topsp 22233 df-bases 22247 df-cld 22321 df-ntr 22322 df-cls 22323 df-nei 22400 df-lp 22438 df-perf 22439 df-cn 22529 df-cnp 22530 df-haus 22617 df-cmp 22689 df-tx 22864 df-hmeo 23057 df-fil 23148 df-fm 23240 df-flim 23241 df-flf 23242 df-xms 23624 df-ms 23625 df-tms 23626 df-cncf 24192 df-limc 25181 df-dv 25182 df-log 25863 df-cxp 25864 |
This theorem is referenced by: divsqrtsumo1 26284 dchrisum0lem2a 26816 |
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